To approximate the volume with 8 boxes, we have to split up the interval of integration for each variable into 2 subintervals, [0, 1] and [1, 2]. Each box will have midpoint [tex]m_{i,j,k}[/tex] that is one of all the possible 3-tuples with coordinates either 1/2 or 3/2. That is, we're sampling [tex]f(x,y,z)=\cos(xyz)[/tex] at the 8 points,
(1/2, 1/2, 1/2)
(1/2, 1/2, 3/2)
(1/2, 3/2, 1/2)
(3/2, 1/2, 1/2)
(1/2, 3/2, 3/2)
(3/2, 1/2, 3/2)
(3/2, 3/2, 1/2)
(3/2, 3/2, 3/2)
which are captured by the sequence
[tex]m_{i,j,k}=\left(\dfrac{2i-1}2,\dfrac{2j-1}2,\dfrac{2k-1}2\right)[/tex]
with each of [tex]i,j,k[/tex] being either 1 or 2.
Then the integral of [tex]f(x,y,z)[/tex] over [tex]B[/tex] is approximated by the Riemann sum,
[tex]\displaystyle\iiint_B\cos(xyz)\,\mathrm dV\approx\sum_{i=1}^2\sum_{j=1}^2\sum_{k=1}^2\cos m_{i,j,k}\left(\frac{2-0}2\right)^2[/tex]
[tex]=\displaystyle\sum_{i=1}^2\sum_{j=1}^2\sum_{k=1}^2\cos\frac{(2i-1)(2j-1)(2k-1)}8[/tex]
[tex]=\cos\dfrac18+3\cos\dfrac38+3\cos\dfrac98+\cos\dfrac{27}8\approx\boxed{4.104}[/tex]
(compare to the actual value of about 4.159)
The Midpoint Rule for triple integrals approximates the integral of a function by taking the function's value at the midpoint of sub-boxes into which the domain is divided. In this problem, B is divided into eight 1x1x1 sub-boxes, and cos(xyz) is evaluated at each sub-box's center.
Explanation:The Midpoint Rule for triple integrals is a method for approximating the value of a triple integral over a box B by dividing the box into smaller sub-boxes and evaluating the function at the midpoint of each sub-box.
For the given integral of cos(xyz) over the box B defined by the ranges 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, and 0 ≤ z ≤ 2, we are asked to divide B into eight sub-boxes of equal size.
This involves calculating the midpoints for each sub-box (xi, yj, zk) and then evaluating the function at these points.
The size of each sub-box would be 1 x 1 x 1 since the original box dimensions are 2 x 2 x 2.
The midpoints for the sub-boxes will be at (0.5, 0.5, 0.5), (0.5, 0.5, 1.5), (0.5, 1.5, 0.5), (0.5, 1.5, 1.5), (1.5, 0.5, 0.5), (1.5, 0.5, 1.5), (1.5, 1.5, 0.5), and (1.5, 1.5, 1.5).
The volume of each sub-box is 1, so the Riemann sum becomes the sum of cos(xyz) evaluated at these midpoints times the volume of each sub-box.
After calculating the Riemann sum, it provides an approximate value for the triple integral.
The actual calculation would require substituting each midpoint into the function cos(xyz) to get the respective value for each sub-box, then summing these values and finally multiplying by the volume of a single sub-box to get an approximation for the entire integral.
this is formula manipulation, I'd appreciate if the steps were provided. only respond if you know how to get the answer, thank you
Answer:
r = 3V/(2πh²)h = 3V/b²r = 25/π cm ≈ 7.9577 cmw = 15 cmStep-by-step explanation:
1. Multiply both sides of the equation by the reciprocal of the coefficient of r.
[tex]V\cdot\dfrac{3}{2\pi h^2}=r\\\\r=\dfrac{3V}{2\pi h^2}[/tex]
__
2. Multiply both sides of the equation by the reciprocal of the coefficient of h.
[tex]V\cdot\dfrac{3}{b^2}=h\\\\h=\dfrac{3V}{b^2}[/tex]
__
3. Solve the circumference formula for r, then substitute the given information.
[tex]C=2\pi r\\\\r=\dfrac{C}{2\pi}\qquad\text{divide by the coefficient of r}\\\\r=\dfrac{50\,\text{cm}}{2\pi}=\dfrac{25}{\pi}\,\text{cm}\approx 7.9577\,\text{cm}[/tex]
__
4. Solve the perimeter formula for width, the substitute the given information and do the arithmetic.
[tex]P=2(L+W)\\\\\dfrac{P}{2}=L+W\qquad\text{divide by 2}\\\\\dfrac{P}{2}-L=W\qquad\text{subtract L}\\\\\dfrac{40\,\text{cm}}{2}-5\,\text{cm}=W=15\,\text{cm}[/tex]
_____
In general, solving for a particular variable involves "undoing" what has been done to the variable, usually in the reverse order. In part 4, the variable W has L added and the sum is multiplied by 2. We "undo" those operations, last operation first, by dividing by 2 and subtracting L.
The properties of equality say you can do what you like to an equation as long as you do the same thing to both sides of the equation. So, when we say "divide by 2", we mean "divide both sides of the equation by 2." Likewise, "subtract L" means "subtract L from both sides of the equation."
In a recent survey, 40 percent indicated chocolate was their favorite flavor of ice cream. Suppose we select a sample of twelve people and ask them to name their favorite flavor of ice cream. a. How many of those in the sample would you expect to name chocolate?
When you have a sample of twelve people and 40 percent of them prefer chocolate ice cream, on average you would expect about 4.8 people to pick chocolate as their favorite flavor. Depending on how you interpret 'expect', this could mean you expect 4 or you 'expect' 5 people.
Explanation:To calculate the expected number of people who prefer chocolate ice cream, we would multiply the total sample size by the percentage that indicated chocolate as their favorite flavor. In this case, the sample size is twelve people, and the percentage that prefers chocolate is 40 percent (or 0.4 when expressed as a decimal).
So, we calculate: 12 * 0.4 = 4.8
However, since we cannot have a fraction of a person, we would either round down to 4 or round up to 5. It depends on what you understand by 'expect'. If you expect at least some amount, then you expect 5 people (because you can't get 4.8 people, you need at least 5 to cover the 4.8). But if you understand 'expect' as an average, then you expect 4 people because fractions are permissible in averages.
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The correct answer is:
Expected number of people who would name chocolate as their favorite flavor is 5.
To solve this problem, we will calculate the expected number of people who would name chocolate as their favorite flavor out of a sample of twelve people, given that 40 percent indicated chocolate was their favorite flavor.
First, we need to determine what 40 percent of twelve people is. To do this, we convert the percentage to a decimal by dividing by 100, which gives us 0.40. Then we multiply this decimal by the number of people in the sample:
[tex]\[ \text{Expected number of people} = 0.40 \times 12 \] \[ \text{Expected number of people} = 4.8 \][/tex]
Since we cannot have a fraction of a person, the expected number is 4.8 people. Rounding off the answer we get 5 as expected number of people.
According to a published study, 13% of men has been involved in a minor traffic accident. Suppose we have randomly and independently sampled 400 and asked each whether he has been involved in a minor traffic accident. How many of the sampled men do we expect to have been involved in a minor traffic accident?
Answer: There are 52 men we can expect to have been involved in a minor traffic accident.
Step-by-step explanation:
Since we have given that
Percentage of men has been involved in a minor traffic accident = 13%
Number of men surveyed = 400
So, Number of sample men we can expect to have been involved in a minor traffic accident is given by
[tex]\dfrac{13}{100}\times 400\\\\=0.13\times 400\\\\=52[/tex]
Hence, there are 52 men we can expect to have been involved in a minor traffic accident.
The population of a town with a 20162016 population of 87 comma 00087,000 grows at a rate of 1.91.9% per year. a. Find the rate constant k and use it to devise an exponential growth function that fits the given data. b. In what year will the population reach 145 comma 000145,000?
Answer:
Given,
The initial population, P = 87,000,
Annual rate of increasing, r = 1.9 %,
a) Thus, the population after t years,
[tex]A=P(1+\frac{r}{100})^t[/tex]
[tex]A=87000(1+\frac{1.9}{100})^{t}[/tex]
[tex]=87000(1+0.019)^t[/tex]
[tex]=87000(1.019)^t[/tex]
[tex]87000(1.019)^t=87000e^{kt}[/tex]
Where, k is the rate constant,
By comparing,
[tex]\implies k=log(1.019)=0.00817418400643\approx 0.00817[/tex]
Hence, the approximate value of k is 0.00817.
And, the exponential growth function would be,
[tex]A=87000e^{0.00817t}[/tex]
b) If A = 145,000,
[tex]145000=87000(1.019)^t[/tex]
By using graphing calculator,
We get,
[tex]t=27.14\approx 27[/tex]
The year after 27 years from 2016 is 2043.
Hence, in 2043 the population reach 145,000.
The rate constant k in the given exponential growth problem is 1.9% per year. The exponential growth function for this problem is P = 87,000 * e^(0.019t). The year in which the population reaches 145,000 can be found by solving the equation 145,000 = 87,000 * e^(0.019t) for t, which will be counted from 2016.
Explanation:The given problem is an example of an exponential growth problem. The formula for exponential growth is P = P0 * e^(kt), where P is the future population, P0 is the initial population (87,000 in this case), e is the natural base (approximately equal to 2.71828), k is the growth rate (1.9% or 0.019 in decimal), and t is the time (in years).
a. Finding the rate constant k
The rate constant k is given in the problem as 1.9% per year or 0.019 per year when expressed in decimal form.
The exponential growth function that fits the given data would therefore be P = 87,000 * e^(0.019t).
b. When will the population reach 145,000?
To find out when the population will reach 145,000, we need to set P equal to 145,000 and solve for t, giving us the equation 145,000 = 87,000 * e^(0.019t). Solving this equation using logarithms will yield the year, which will be counted from the base year 2016.
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Find the volume of the prism.
The correct option is B.
Any prism volume is V = BH where B is an area of the base and H is the height of the prism, so find the area of the base by B = 1/2 h(b1+b2), then multiply by the height of the prism.
How to Find the volume of the prism?
The triangles on the two ends are congruent.
So, to find the prism area, it is enough to take the area of one of the triangles and multiply it by the prism depth, which is 20 units.
Use the Pythagorean theorem to find the height of the triangle.
5^2 - 3^2 = 4^2, so the height is 4.
Multiply half times height times base: 1/2 x 4 x 3 = 6.
Multiply that by the depth, 20.
6x20=120 units^3.
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"The average starting salary of individuals with a master's degree in statistics is normally distributed with a mean of $48,000 and a standard deviation of $6,000. What is the probability that a randomly selected individual with a master's in statistics will get a starting salary of at least $55,000?
Answer:
12.1%
Step-by-step explanation:
First calculate the z-score:
z = (x − μ) / σ
z = (55000 − 48000) / 6000
z = 1.17
Look up in a z-score table.
P(z>1.17) = 1 − 0.8790
P(z>1.17) = 0.1210
There's a 12.1% probability that a randomly selected individual with a master's in statistics will have a starting salary of at least $55,000.
The probability that a randomly selected individual with a Master's in statistics will get a starting salary of at least $55,000 is 12.17%.
Explanation:The problem you're dealing with involves the normal distribution which is a type of continuous probability distribution for a real-valued random variable. Given that we have a mean (μ) of $48,000, and a standard deviation (σ) of $6,000, we are asked to find the probability that a randomly selected individual with a master's in statistics will get a starting salary of at least $55,000.
This involves calculating a 'Z-score', which tells us how many standard deviations an element is from the mean. To find it, you would subtract the mean from the value of interest and divide by the standard deviation.
So, in this case, the Z-score would be: Z = (55000 - 48000) / 6000 = 1.167. Using standard Z-score tables, this corresponds to a probability of 0.8783.
However, as we are interested in the probability of a salary being at least $55,000, we need to subtract this value from 1 to get the final probability. So, the answer is: 1 - 0.8783 = 0.1217 or 12.17%.
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jane bought a case of wine for $345. She kept three bottles of wine for herself and sold the rest for $480, making a profit of $17 on each bottle of wine. How many botles of wine were in the case?
Answer: 31.23
Step-by-step explanation: Jane kept 3 bottles for her self and then sold the rest for 17 $ each and the amount of money she made off of selling the rest of the bottles was 480$ and if you divide 480 by 17 you get 28.23 bottles sold and the extra 3 she kept for herself I am not 100% sure.
Jane bought 18 bottles of wine in total. She kept 3 for herself and sold 15, making a profit of $17 on each of the 15 bottles.
Jane bought a case of wine for $345, kept three bottles for herself, and sold the rest for $480, making a profit of $17 on each bottle sold. There were 18 bottles in the case.
The correct approach is to find the factors of $345 and $480 that are 3 bottles apart since Jane kept three bottles.
The factors of $345 are 1, 3, 5, 7, 15, 21, 23, 35, 45, 49, 69, 77, 115, 147, and 345.
The factors of $480 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 80, 96, 120, 160, 240, and 480.
The common factors are 1, 3, 5, and 15. Since Jane made a profit on each bottle sold, the number of bottles sold must be more than the number of bottles she kept. Therefore, the number of bottles Jane could have bought and sold must be a factor of $480 that is 3 more than a factor of $345.
The only factor of $480 that is 3 more than a factor of $345 is 18, since 15 (a factor of $345) + 3 = 18 (a factor of $480).
Consider the set of differences, denoted with d, between two dependent sets: 84, 85, 83, 63, 61, 100, 98. Find the sample standard deviation sd and round it to the nearest tenth.
Answer:
The sample standard deviation is 15.3.
Step-by-step explanation:
Given data items,
84, 85, 83, 63, 61, 100, 98,
Number of data items, N = 7,
Let x represents the data item,
Mean of the data points,
[tex]\bar{x}=\frac{84+85+83+63+61+100+98}{7}[/tex]
[tex]=82[/tex]
Hence, sample standard deviation would be,
[tex]\sigma= \sqrt{\frac{1}{N-1}\sum_{i=1}^{N} (x_i-\bar{x})^2}[/tex]
[tex]=\sqrt{\frac{1}{6}\sum_{i=1}^{7} (x_i-82)^2}[/tex]
[tex]=\sqrt{\frac{1}{6}\times 1396}[/tex]
[tex]=\sqrt{232.666666667}[/tex]
[tex]=15.2534149182[/tex]
[tex]\approx 15.3[/tex]
The sample standard deviation of the dataset: 84, 85, 83, 63, 61, 100, 98 is approximately 15.3 when rounded to the nearest tenth.
Explanation:To find the sample standard deviation of the given set, we first need to calculate the mean of the data set. Then, each number in the data set should be subtracted from the mean, and the results squared. These squared differences should be summed and divided by the number of data values minus one, which gives the variance. Taking the square root of the variance gives the sample standard deviation.
Let's do this step by step for the given dataset: 84, 85, 83, 63, 61, 100, 98.
Find the mean (average): (84+85+83+63+61+100+98) / 7 = 82 Subtract the mean from each number and square the result: (84-82)^2, (85-82)^2, ..., (98-82)^2. Sum these squared differences: 4+9+1+361+441+324+256 = 1396. Divide this sum by the number of data values minus one (n - 1): 1396 / (7 - 1) = 232.67 (This is the variance). Take the square root of the variance to get the sample standard deviation: √232.67 = 15.3 (rounded to the nearest tenth). Learn more about Sample Standard Deviation here:https://brainly.com/question/30952578
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Find a bijection from (0,1] to (0,1)
Math: Analysis and proof
Answer with explanation:
The meaning of bijection for two sets X and Y is each and every element of X is uniquely related with element of set Y and when you take the inverse mapping every element of set Y is uniquely related with each and every element of Set X.
The two sets given are
X=(0,1]---------Semi open or Semi closed Set
Y=(0,1)---------Closed set
Between any two real numbers there are infinite number of real numbers.So cardinal number of both the sets is infinite.
There can be infinite bijection between these two sets as both sets have infinite number of elements.
X Y
0.1 ------------- 0.01
0.2 ------- 0.02
0.3 -------- 0.03
0.4 --------- 0.04
-------------------------
------------------------------
-------------------------------
--------------------------------------
One example of a bijection from [tex]\((0,1]\) to \((0,1)\)[/tex] is the function [tex]\(f(x) = \frac{x}{1 + x}\)[/tex]. This function is injective (one-to-one) and surjective (onto), making it a bijection.
To establish a bijection from [tex]\((0,1]\) to \((0,1)\)[/tex], consider the function [tex]\(f(x) = \frac{x}{1 + x}\).[/tex]
1. Injectivity (One-to-One): Assume [tex]\(f(x_1) = f(x_2)\)[/tex] for some [tex]\(x_1, x_2 \in (0,1]\)[/tex]. By solving the equation [tex]\(\frac{x_1}{1 + x_1} = \frac{x_2}{1 + x_2}\)[/tex], you find [tex]\(x_1 = x_2\)[/tex], showing that distinct elements in the domain map to distinct elements in the codomain.
2. Surjectivity (Onto): For any y in the codomain [tex]\((0,1)\)[/tex], solve [tex]\(f(x) = y\) for \(x\)[/tex]. This results in [tex]\(x = \frac{y}{1 - y}\)[/tex], which is well-defined for [tex]\(y \in (0,1)\)[/tex]. Therefore, every element in the codomain has a preimage in the domain.
Hence, [tex]\(f(x)\)[/tex] is a bijection from [tex]\((0,1]\) to \((0,1)\)[/tex].
The area under the normal curve between the mean and a score for which z= - 1.20 is
a. .2563
b. .7563
c. .1138
d. .3862
e. .3849
Answer: e. 0.3849
Step-by-step explanation:
We know that the mean lies exactly at the middle of the normal curve .
The z-score of the mean value is 0.
Also According to the standard normal probability table, the probability value of z=0 is P(z<0)=0.5.
And the probability value of z=-1.20 is P(z<-1.2) =0.1150697.
Now, the area under the normal curve between the mean and a score for which z= - 1.20 is given by :-
[tex]P(z<0)-P(z<-1.2)=0.5-0.1150697=0.3849303\approx0.3849[/tex]
Hence, the area under the normal curve between the mean and a score for which z= - 1.20 is 0.3849 .
certain electronics manufacturer found that the marginal cost C to produce x DVD/Blu- ray players can be found using the equation C = 0.03 x 2 − 8 x + 700 . If the marginal cost were $522, how many DVD/Blu-ray players were produced?
Answer:
242.
Step-by-step explanation:
C = 0.03 x 2 − 8 x + 700
When the marginal cost = $522 we have:
522 = 0.03 x 2 − 8 x + 700
0.03x^2 - 8x + 700 - 522 = 0
0.03x^2 - 8x + 178 = 0
Using the quadratic formula:
x = [ -(-8) +/- sqrt( 64 - 4 * 0.03 * 178)] / 0.06
= 8 +/- sqrt42.64 / 0.06
= 242 players.
242 DVD/Blu-ray players were produced
Marginal cost is the change in cost as a result of producing additional number of units.
Given that the marginal cost is:
C = 0.03x² - 8x + 700
Therefore for a marginal cost of $522, the number of DVD players is:
522 = 0.03x² - 8x + 700
0.03x² - 8x + 178 = 0
x = 24 and x = 242
Therefore 242 DVD/Blu-ray players were produced
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State the domain and the range of each relation. Then determine whether the relation is a function.
(4,23)
(3,-2)
(-6,5)
(4,6)
Answer:
[tex]Domain:[/tex]{[tex]-6,3,4[/tex]}
[tex]Range:[/tex]{[tex]-2,5,6,23[/tex]}
The relation is not a function.
Step-by-step explanation:
By definition, a relation is a function if each input value has only one output value.
Given the relation:
(4,23)
(3,-2)
(-6,5)
(4,6)
The domain is the set of the x-coordinates of each ordered pair (You do not need to write 4 twice):
[tex]Domain:[/tex]{[tex]-6,3,4[/tex]}
The range is the set of the y-coordinates of each ordered pair :
[tex]Range:[/tex]{[tex]-2,5,6,23[/tex]}
Since the input value 4 has two different output values (23 and 6), the relation is not a function.
Answer:
See below.
Step-by-step explanation:
The Domain = {-6, 3, 4}.
The Range = {-2, 5, 6, 23}.
The relation is NOT a function because 4 in the domain maps to 6 and 23. In a function an element in the domain must map to one element only in the range.
Using the Extended Euclidean Algorithm, find integers x and y such that 26x + 9y = 1.
Answer:
The solution is: x=-1 and y=3.
Step-by-step explanation:
Let's find the solution, but first let's remember the following:
A / B = C + R where:
A=dividend, B=divisor, C=quotient, and R=remainder. This can be express as follows:
A = (C * B) + R, which is the structure we are going to use next.
Using the Euclidian Algorithm we need to find the highst common factor (HCF) between the coefficients from your equation, this means:
The original equation: 26x+9y=1 is in the form Ax+By=C, so A=26, B=9 and C=1.
We need to find the HCF of 'A' and 'B'. Using the Euclidan Algorithm (EA), so we have:
A = (C * B) + R, using our values:
26 = (2 * 9) + 8, look that the divisor (B) is 9 and the remainder (R) is 8.
Now using the (EA) we divide the divisor (B=9) by the obtained remainder (R=8). And we do the same for each obtained result until the las remainder (R) is equal to 0, like this:
A = (C * B) + R
9 = (8 * 1) + 1
using the divisor (B=8) and the remainder (R=1) we obtain:
A = (C * B) + R
8 = (1 * 8) + 0, look that the remainder is now 0, so in summary we can use the method as follows:
26 = (2 * 9) + 8
9 = (8 * 1) + 1
8 = (1 * 8) + 0; and this equations are the ones we are going to use in order to find a solution.
Next step is to use the equation before R=0, so:
9 = (8 * 1) + 1, which is:
9 - (8 * 1) = 1; but if you consider the first obtained equation: 26 = (2 * 9) + 8, we can write:
26 - (2 * 9) = 8, and we can use this expression in the previous one, so:
9 - (8 * 1) = 1, is:
9 - ((26 - (2 * 9)) * 1) = 1, simplifying:
9 - 26 + (2*9) = 1
9 - 26 + (9 + 9) = 1
-26 + (9 + 9 + 9) = 1
-26 + (3 * 9) = 1
26*(-1) + 9*(3) = 1; if you compare the last expression with the original equation, which is: 26x+9y=1, you can see the similarity, where x=-1 and y=3.
So, the solution is: x=-1 and y=3.
The length of a rectangle exceeds its width by 8 inches. The perimeter is 80 inches. Find the length and the width of the rectangle.
Answer:
The width is 16 inches while the length is 24 inches.
Step-by-step explanation:
We are given that the length (L) is 8 more than it's width (W).
The equation for this is: L=8+W.
We are given the perimeter is 80.
The equation for this is 2L+2W=80.
Each term here is even so I'm going to divide both sides by 2: L+W=40.
My system to solve is:
L=8+W
L+W=40
I'm going to plug the first equation into the second giving me:
(8+W)+W=40
Drop the (), don't really need:
8+W+W=40
Combine like terms:
8+2W=40
Subtract 8 on both sides:
2W=32
Divide both sides by 2:
W=32/2
Simplify:
W=16
So if W=16 and L=8+W then L=8+16=24.
The width is 16 inches while the length is 24 inches.
Perimeter is the sum of the length of the sides used to make the given figure. The length and the width of the rectangle are 24 inches and 16 inches, respectively.
What is the perimeter?Perimeter is the sum of the length of the sides used to make the given figure.
Let the width of the rectangle be represented by x.
Given that the length of a rectangle exceeds its width by 8 inches. Therefore, the length and the width of a rectangle can be written as,
Width = x inches
Length = x inches + 8 inches
Now, as it is given that the perimeter of the rectangle is 80 inches. Therefore, the perimeter of the given rectangle can be written as,
Perimeter of rectangle = 2(Length + Width)
80 inches = 2[(x+8) + x]
80 = 2(x + 8 + x)
80 = 2(2x + 8)
80/2 = 2x + 8
40 = 2x + 8
40 - 8 = 2x
32 = 2x
x = 32/2
x = 16 inches
Further, the length and the width of the rectangle can be written as,
Width = x = 16 inches
Length = x + 8 = 16 + 8 = 24 inches
Hence, the length and the width of the rectangle are 24 inches and 16 inches, respectively.
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How many integers between 10000 and 99999, inclusive, are divisible by 3 or
5 or 7?
Answer: Hence, there are approximately 48884 integers are divisible by 3 or 5 or 7.
Step-by-step explanation:
Since we have given that
Integers between 10000 and 99999 = 99999-10000+1=90000
n( divisible by 3) = [tex]\dfrac{90000}{3}=30000[/tex]
n( divisible by 5) = [tex]\dfrac{90000}{5}=18000[/tex]
n( divisible by 7) = [tex]\dfrac{90000}{7}=12857.14[/tex]
n( divisible by 3 and 5) = n(3∩5)=[tex]\dfrac{90000}{15}=6000[/tex]
n( divisible by 5 and 7) = n(5∩7) = [tex]\dfrac{90000}{35}=2571.42[/tex]
n( divisible by 3 and 7) = n(3∩7) = [tex]\dfrac{90000}{21}=4285.71[/tex]
n( divisible by 3,5 and 7) = n(3∩5∩7) = [tex]\dfrac{90000}{105}=857.14[/tex]
As we know the formula,
n(3∪5∪7)=n(3)+n(5)+n(7)-n(3∩5)-n(5∩7)-n(3∩7)+n(3∩5∩7)
[tex]=30000+18000+12857.14-6000-2571.42-4258.71+857.14\\\\=48884.15[/tex]
Hence, there are approximately 48884 integers are divisible by 3 or 5 or 7.
You sell tickets at school for fundraisers. You sold car wash tickets, silly string fight tickets and dance tickets – for a total of 380 tickets sold. The car wash tickets were $5 each, the silly sting fight tickets were $3 each and the dance tickets were $10 each. If you sold twice as many silly string tickets as car wash tickets, and you have $1460 total, how many dance tickets were sold?
Answer:
11 dance tickets were sold. ( approx )
Step-by-step explanation:
Let x represents the number of car wash tickets, y represents the number of silly string fight tickets and z represents number of dance tickets,
Total tickets = 380,
⇒ x + y + z = 380 -----(1),
The car wash tickets were $5 each, the silly string fight tickets were $3 each and the dance tickets were $10 each, also total cost is $1460,
⇒ 5x + 3y + 10z = 1460 ----(2)
Also, there are twice as many silly string tickets as car wash tickets,
⇒ y = 2x -----(3)
From equation (1) and (2),
x + 2x + z = 380 ⇒ 3x + z = 380 -----(4)
5x + 3(2x) + 10z = 1460 ⇒ 11x + 10z = 1460 ------(5)
For finding the value of z,
3 × equation (5) - 11 × equation (4),
We get,
30z - 11z = 4380 - 4180
19z = 200
z = 10.5263157895 ≈ 11
Hence, 11 dance tickets were sold.
Determine whether the series is convergent or divergent. Hint: First use partial fraction decomposition. ∞ 8 (n−1)(n+1) n = 3 convergent divergent Correct: Your answer is correct. If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)
I guess the series is
[tex]\displaystyle\sum_{n=3}^\infty\frac8{(n-1)(n+1)}[/tex]
Splitting into partial fractions gives
[tex]\dfrac8{(n-1)(n+1)}=\dfrac4{n-1}-\dfrac4{n+1}[/tex]
The series telescopes, which we can see by considering the [tex]k[/tex]-th partial sum:
[tex]\displaystyle\sum_{n=3}^k\frac8{(n-1)(n+1)}=(2-1)+\left(\frac43-\frac45\right)+\cdots+\left(\frac4{k-2}-\frac4k\right)+\left(\frac4{k-1}-\frac4{k+1}\right)[/tex]
(feel free to write out more terms to convince yourself that the following is true)
[tex]\displaystyle\sum_{n=3}^k\frac8{(n-1)(n+1)}=2+\frac43-\frac4k-\frac4{k+1}[/tex]
Then as [tex]k\to\infty[/tex], we end up with
[tex]\displaystyle\lim_{k\to\infty}\sum_{n=3}^k\frac8{(n-1)(n+1)}=\boxed{\frac{10}3}[/tex]
The series is convergent with a sum of -1.143
Explanation:The given series is:
∞ 8 (n-1)(n+1) / n = 3
First, we can use partial fraction decomposition to simplify the series. Factoring (n-1)(n+1) gives us [tex]n^2[/tex] - 1. So, the series becomes:
∞ 8 ([tex]n^2[/tex] - 1) / n = 3
Next, we can rewrite the series as two separate series:
∞ 8 [tex]n^2[/tex] / n - 8 / n = 3
Simplifying each series:
∞ 8n - 8 / n = 8∞ n³ - 1∞
The first series is a geometric series with a common ratio of 8 and a first term of 8. Using the formula for the sum of a geometric series, we can find its sum:
S = a / (1-r)
Substituting the values:
S = 8 / (1-8) = 8 / -7 = -1.143
Therefore, the series is convergent, and its sum is -1.143.
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HELP! Need quick - Determine whether the data distribution appears to be positively skewed, negatively skewed, or symmetric.
The table below shows the average employee salary (in thousands) at S&S, Inc for the last five years.
A) negatively skewed
B) symmetric
C) positively skewed
Answer:
A. negatively skewed
Step-by-step explanation:
Plot the values of salary against years
You will notice that;
The graph has a long left tailThe tail is longer towards the negative direction of the number lineThis mean that this graph is a negative skewed graph/left-skewed distribution
In the attached sketch of the plot, join the points with a smooth curve and observe the above mentioned properties.
The x-axis scale ranges from 2000 to 2015 where 0=2000, 1=2001,2=2002......,15=2015
Hope this will give you a visual picture of the negatively- skewed distribution
|3 - 10| - (12/4 +2)^2
Answer:
-18
Step-by-step explanation:
Simplify the following:
abs(3 - 10) - (12/4 + 2)^2
The gcd of 12 and 4 is 4, so 12/4 = (4×3)/(4×1) = 4/4×3 = 3:
abs(3 - 10) - (3 + 2)^2
3 + 2 = 5:
abs(3 - 10) - 5^2
3 - 10 = -7:
abs(-7) - 5^2
5^2 = 25:
abs(-7) - 25
Since -7<=0, then abs(-7) = 7:
7 - 25
7 - 25 = -18:
Answer: -18
For this case we must resolve the expression:
[tex]| 3-10 | - (\frac {12} {4}+2) ^ 2[/tex]
We have the following definitions:
Different signs are subtracted and the sign of the major is placed.
Absolute value definition:[tex]| -a | = a[/tex]
Resolving we have:
[tex]| 3-10 | = | -7 | = 7[/tex]
[tex](\frac {12} {4}+2) ^ 2 = (3+2) ^ 2 = 5 ^ 2 = 25[/tex]
Then, rewriting the expression:
[tex]7-25 = -18[/tex]
Answer:
-18
The equation of a circle is given below. Identify the radius and center.
x^2 + y^2 + 6x -8y +21 = 0
[tex](x-a)^2+(y-b)^2=r^2[/tex]
center - [tex](a,b)[/tex]
[tex]x^2 + y^2 + 6x -8y +21 = 0 \\x^2+6x+9+y^2-8y+16-4=0\\(x+3)^2+(y-4)^2=4[/tex]
center - [tex](-3,4)[/tex]
[tex]r=2[/tex]
Answer:
The center is: [tex](-3, 4)[/tex] and the radius is [tex]r=2[/tex]
Step-by-step explanation:
The general equation of a circle has the following formula:
[tex](x-h)^2 + (y-k)^2 = r^2\\[/tex]
Where r is the radius of the circle and (h, k) is the center of the circle
In this case we have the following equation
[tex]x^2 + y^2 + 6x -8y +21 = 0[/tex]
To find the radius and the center of this cicunference we must rewrite the equation in the general form of a circumference completing the Square
[tex](x^2 + 6x)+ (y^2 -8y) +21 = 0\\\\(x^2 + 6x+9)+ (y^2 -8y+16) +21 = 9+16\\\\(x^2 + 6x+9)+ (y^2 -8y+16) = 9+16-21\\\\(x+3)^2+ (y-4)^2 = 4\\\\(x+3)^2+ (y-4)^2 = 2^2[/tex]
Then the center is: [tex](-3, 4)[/tex] and the radius is [tex]r=2[/tex]
Enter your PIN: The technology company DataGenetics suggests that % of all four-digit personal identification numbers, or PIN codes, have a repeating digits format such as . Assuming this to be true, if the PIN codes of seven people are selected at random, what is the probability that at least one of them will have repeating digits? Round your answer to four decimal places.
Answer: 0.9917
Step-by-step explanation:
If repetition is allowed , then the total number of possible four digits pin codes = [tex]10^4=10,000[/tex]
Number of ways to make for digit code without repetition of digits =
[tex]10\times9\times8\times7=5040[/tex]
Number of ways to make for digit codes having repetition =
[tex]10,000-5040=4960[/tex]
Probability that a person has pin code that has repetition:-
[tex]\dfrac{4960}{10,000}=0.496[/tex]
Let x be number of pin codes with repeating digits.
If the PIN codes of seven people are selected at random, then the probability that at least one of them will have repeating digits:-
[tex]P(x\geq1)=1-(P(0))\\\\=1-(^7C_0(0.496)^0(1-0.496)^7)[/tex] (By Binomial distribution)
[tex]=1-((0.496)^0(0.504)^7)=0.991739358875\approx0.9917[/tex]
Hence, the probability that at least one of them will have repeating digits = 0.9917
Imagine a regression line that relates y = average systolic blood pressure to x = age. The average blood pressure for people 40 years old is 110, while for those 50 years old the average is 130. (a) What is the slope of the regression line? (Round the answer to one decimal place.)
Answer:
2.0
Step-by-step explanation:
x = age
y = average systolic blood pressure
x₁ = 40 years old, y₁ = 110 systolic blood pressure
x₂ = 50 years old, y₂ = 130 systolic blood pressure
[tex]\text{Slope of a line}=m=\frac{y_2-y_1}{x_2-x_1}\\\Rightarrow m= \frac{130-110}{50-40}\\\Rightarrow\frac{20}{10}\\\Rightarrow m= 2.0[/tex]
∴ Slope of the regression line is 2.0
Final answer:
The slope of the regression line relating average systolic blood pressure to age is 2.0 when rounded to one decimal place.
Explanation:
To find the slope of the regression line relating average systolic blood pressure (y) to age (x), we can use the two points provided (x1, y1) = (40, 110) and (x2, y2) = (50, 130). The slope (m) of a line passing through two points is calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the given values:
m = (130 - 110) / (50 - 40) = 20 / 10 = 2
Therefore, the slope of the regression line is 2.0 when rounded to one decimal place.
Retail prices in a department store are obtained by marking up the wholesale price by 60 %. That is, retail price is obtained by adding 60 % of the wholesale price to the wholesale price.
(A) What is the retail price of a suit if the wholesale price is $200?
(B) What is the wholesale price of a pair of jeans if the retail price is $80?
A) The retail price of the suit would be $320.
B) The wholesale price of the pair of jeans would be $50.
Given that;
Retail prices in a department store are obtained by marking up the wholesale price by 60 %. That is, the retail price is obtained by adding 60 % of the wholesale price to the wholesale price.
(A) Now for the retail price of the suit, we need to add 60% of the wholesale price to the wholesale price itself.
So, if the wholesale price is $200, calculate the retail price as follows:
Retail price = Wholesale price + (60% of Wholesale price)
Retail price = $200 + (0.60 × $200)
Retail price = $200 + $120
Retail price = $320
Therefore, the retail price of the suit would be $320.
(B) Now, for the wholesale price of the pair of jeans, reverse the calculation.
If the retail price is $80 and the markup is 60%, calculate the wholesale price as follows:
Wholesale price = Retail price / (1 + Markup percentage)
Wholesale price = $80 / (1 + 0.60)
Wholesale price = $80 / 1.60
Wholesale price = $50
Hence, the wholesale price of the pair of jeans would be $50.
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The retail price of a suit with a wholesale price of $200 would be $320. To find the wholesale price of a pair of jeans that retails for $80, you calculate the original wholesale price to be $50.
Explanation:To answer question (A), we calculate the retail price by adding 60% of the wholesale price to the wholesale price itself. For a wholesale price of $200, this would be:
Retail Price = Wholesale Price + (0.60 × Wholesale Price)
Retail Price = $200 + (0.60 × $200)
Retail Price = $200 + $120
Retail Price = $320
For question (B), we find the wholesale price of a pair of jeans if the retail price is $80 by letting W represent the wholesale price and solving for W:
Retail Price = W + (0.60 × W)
$80 = W + (0.60 × W)
$80 = 1.60W
W = $80 / 1.60
W = $50
Several psychology students are unprepared for a surprise true/false test with 18 questions, and all of their answers are guesses. a. Find the mean and standard deviation for the number of correct answers for such students. b. Would it be unusual for a student to pass by guessing (which requires getting at least 14 correct answers)? Why or why not?
In a true/false test where answers are guessed, the mean number of correct answers out of 18 would be 9 (50%) and the standard deviation would be 3. Passing by guessing with a minimum of 14 correct answers would be unusual as it is significantly above the mean and more than a single standard deviation away.
Explanation:This question is about applying the binary distribution, specifically dealing with a true/false test. Before diving into the question, let's understand what we're working with. In a true/false test where students are guessing, the probability of a correct answer (p) is 0.5, as the expected value (mean) is np (number of trials times the probability of success) and the standard deviation is sqrt(np(1-p)).
In this scenario, for 18 questions, the mean (expected value) would be 18*0.5 = 9 and the standard deviation would be sqrt(18*0.5*0.5) = 3. Therefore, the mean is 9 and the standard deviation is 3.
For a student to 'pass' by guessing, they would need at least 14 correct answers. This number of successes is much greater than our mean expectation of 9, and it's more than a single standard deviation away. Therefore, it would be unusual for a student to pass the test by simply guessing.
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You just purchased two coins at a price of $670 each. Because one of the coins is more collectible, you believe that its value will increase at a rate of 7.1 percent per year, while you believe the second coin will only increase at 6.5 percent per year. If you are correct, how much more will the first coin be worth in 15 years?
Answer:
The value of first coin will be $151.51 more than second coin in 15 years.
Step-by-step explanation:
You have just purchased two coins at a price of $670 each.
You believe that first coin's value will increase at a rate of 7.1% and second coin's value 6.5% per year.
We have to calculate the first coin's value after 15 years by using the formula
[tex]A=P(1+\frac{r}{100})^{n}[/tex]
Where A = Future value
P = Present value
r = rate of interest
n = time in years
Now we put the values
[tex]A=670(1+\frac{7.1}{100})^{15}[/tex]
[tex]A=670(1+0.071)^{15}[/tex]
[tex]A=670(1.071)^{15}[/tex]
A = (670)(2.797964)
A = 1874.635622 ≈ $1874.64
Now we will calculate the value of second coin.
[tex]A=670(1+\frac{6.5}{100})^{15}[/tex]
[tex]A=670(1+0.6.5)^{15}[/tex]
[tex]A=670(1.065)^{15}[/tex]
A = 670 × 2.571841
A = $1723.13
The difference of the value after 15 years = 1874.64 - 1723.13 = $151.51
The value of first coin will be $151.51 more than second coin in 15 years.
To find out how much more the first coin will be worth compared to the second coin in 15 years, we can apply the formula for compound interest. The formula is:
\[ A = P(1 + r)^t \]
where:
- \( A \) is the future value of the investment/loan, including interest
- \( P \) is the principal investment amount (the initial deposit or loan amount)
- \( r \) is the annual interest rate (decimal)
- \( t \) is the number of years the money is invested or borrowed for
We will apply this formula separately for each coin and then find the difference between the two future values.
First, let's calculate the future value of the more collectible coin:
\( P = \$670 \) (the initial value of the coin)
\( r = 7.1\% = 0.071 \) (the annual increase rate converted to decimal)
\( t = 15 \) years
So the future value \( A_1 \) for the first coin is calculated as:
\[ A_1 = \$670 \times (1 + 0.071)^{15} \]
Now, let's calculate the future value of the less collectible coin:
\( P = \$670 \) (the initial value of the coin)
\( r = 6.5\% = 0.065 \) (the annual increase rate converted to decimal)
\( t = 15 \) years
So the future value \( A_2 \) for the second coin is calculated as:
\[ A_2 = \$670 \times (1 + 0.065)^{15} \]
Finally, to find out how much more the first coin will be worth, we subtract the future value of the second coin from the future value of the first coin:
\[ \text{Difference} = A_1 - A_2 \]
Let's compute this step by step.
First, calculate \( A_1 \):
\[ A_1 = \$670 \times (1 + 0.071)^{15} \]
\[ A_1 = \$670 \times (1.071)^{15} \]
\[ A_1 = \$670 \times 2.83844... \] (using a calculator)
\[ A_1 \approx \$1901.75 \] (rounded to two decimal places)
Next, calculate \( A_2 \):
\[ A_2 = \$670 \times (1 + 0.065)^{15} \]
\[ A_2 = \$670 \times (1.065)^{15} \]
\[ A_2 = \$670 \times 2.64716... \] (using a calculator)
\[ A_2 \approx \$1775.63 \] (rounded to two decimal places)
Now calculate the difference:
\[ \text{Difference} = A_1 - A_2 \]
\[ \text{Difference} = \$1901.75 - \$1775.63 \]
\[ \text{Difference} \approx \$126.12 \]
Therefore, if your expectations are correct, in 15 years, the first coin will be worth approximately $126.12 more than the second coin.
Identify the symbols used for each of the following: (a) sample standard deviation; (b) population standard deviation; (c) sample variance; (d) population variance.a. The symbol for sample standard deviation is▼ss squaredsigmasigma squared.b. The symbol for population standard deviation is▼ss squaredsigmasigma squared.c. The symbol for sample variance is▼ss squaredsigmasigma squared.d. The symbol for population variance is▼ss squaredsigmasigma squared.
Answer:
The symbol used for the sample standard deviation is the 's', while the symbol used for the population standard deviation is a "sigma" or σ.
On the other hand, the sample variance comes to be the square of the sample standard deviation: s². Likewise, the the population variance comes to be the square of the population standard deviation which is: σ²
The symbols used for sample standard deviation, population standard deviation, sample variance, and population variance are s, σ (sigma), s², and σ² (sigma squared), respectively.
Explanation:When discussing statistics, there are specific symbols used for different concepts. For the (a) sample standard deviation, the most commonly used symbol is s. In terms of (b) population standard deviation, the symbol commonly used for this is a Greek letter, σ (lowercase sigma).
Then, when shifting the focus from standard deviation to variance, there are different symbols. For (c) sample variance, the symbol used is s², which is the sample standard deviation squared. As for the (d) population variance, this is represented by the square of the population standard deviation, σ² (sigma squared).
The symbols used for each of the following are:
(a) Sample standard deviation: s
(b) Population standard deviation: σ (lowercase sigma)
(c) Sample variance: s²
(d) Population variance: σ² (lowercase sigma squared)
To calculate the standard deviation, you first need to calculate the variance. The variance is the average of the squares of the deviations. The sample standard deviation is represented by s, while the population standard deviation is represented by lowercase Greek letter σ (sigma).
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Complete question
Identify the symbols used for each of the following:
(a) sample standard deviation;
(b) population standard deviation;
(c) sample variance;
(d) population variance.
a. The symbol for sample standard deviation is▼ss squaredsigmasigma squared.
b. The symbol for population standard deviation is▼ss squaredsigmasigma squared.
c. The symbol for sample variance is▼ss squaredsigmasigma squared.
d. The symbol for population variance is▼ss squaredsigmasigma squared.
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A fluid has density 810 kg/m3 and flows with velocity v = z i + y2 j + x2 k, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the cylinder x2 + y2 = 9, 0 ≤ z ≤ 4.
To find the outward flow through the cylinder [tex]\(x^2 + y^2 = 4\), \(0 \leq z \leq 4\),[/tex] integrate the dot product of velocity and surface normal. The outward flux is [tex]\(810 \times 32 \pi\).[/tex]
To calculate the rate of flow outward through the cylinder [tex]\(x^2 + y^2 = 4\), \(0 \leq z \leq 4\),[/tex] let's compute the flux of the given vector field through the cylindrical surface. The outward flux through a closed surface is given by:
[tex]\[\Phi = \int_S \mathb{v} \cdot \mathb{dS}.\][/tex]
Surface Parameterization and Surface Normal.
The cylinder has radius r = 2, so the general parameterization is:
- [tex]\(x = 2 \cos(\theta)\),[/tex]
- [tex]\(y = 2 \sin(\theta)\),[/tex]
- z = z,
where [tex]\(0 \leq \theta \leq 2\pi\) and \(0 \leq z \leq 4\).[/tex]
The outward normal for the cylindrical surface is [tex]\(\mathb{n} = \cos(\theta) \mathb{i} + \sin(\theta) \mathb{j}\).[/tex]
The differential surface element is:
[tex]\[\mathb{dS} = 2 \, d\theta \, dz \, (\cos(\theta) \mathb{i} + \sin(\theta) \mathb{j}).\][/tex]
Dot Product of Velocity and Normal :
The given velocity field is:
[tex]\[\mathb{v} = z \mathb{i} + y^2 \mathb{j} + x^2 \mathb{k}.\][/tex]
The dot product of the velocity with the surface normal is:
[tex]\[\mathb{v} \cdot \mathb{dS} = 2 \, dz \, d\theta \, (z \cos(\theta) + (4 \sin^2(\theta)) \sin(\theta)).\][/tex]
Integrate to Find the Flux :
The flux through the cylindrical surface is given by:
[tex]\[\int_0^4 \int_0^{2\pi} 2 \, (z \cos(\theta) + 4 \sin^2(\theta)) \, dz \, d\theta,[/tex]
Separate and compute the integral:
- The integral of [tex]\(z \cos(\theta)\)[/tex] over[tex]\((0, 2\pi)\)[/tex] is zero (because [tex]\(\cos(\theta)\)[/tex] has symmetric oscillations).
- The integral of [tex]\(4 \sin^2(\theta)\) over \((0, 2\pi)\) is \(2\pi \cdot 4\),[/tex] since [tex]\(\sin^2(\theta) = \frac{1}{2}(1 - \cos(2\theta))\).[/tex]
This results in [tex]\(8\pi\).[/tex]
To compute the flux, multiply by 4 : [tex]\[8\pi \times 4 = 32\pi.\][/tex]
Since the density of the fluid is 810 kg/m³, the outward flux of the fluid through the cylinder, considering the density, is : [tex]\[810 \times 32 \pi.[/tex]
This would be the answer, with the expression [tex]\(810 \times 32 \pi\)[/tex] giving the rate of flow outward through the cylinder.
Complete question : A fluid has density 810 kg/m3 and flows with velocity v = z i + y2 j + x2 k, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the cylinder x2 + y2 = 4, 0 ≤ z ≤ 4.
A computer password system requires 6 DIFFERENT letters (case sensitive) from the 26 alphabets. For example, AAAAAA and aaaaaa are two different passwords. Is this a PERMUTATION or a COMBINATION problem? How many passwords can be created?
Answer: 325
Step-by-step explanation:
Combination is a way to calculate the total outcomes of an event where order of the outcomes does not matter where as a Permutation is a way of arranging the elements of a set into a order or a sequence . Here order matters.
If we want to create password with 6 different letters then order matters.
Hence, we use permutations.
The number of passwords created is given by :-
[tex]^{26}P_6=\dfrac{26!}{2!(26-2)!}\\\\=\dfrac{26\times25\times24!}{2\times24!}=325[/tex]
Hence, the number of passwords created = 325
In a recent year, companies spent a total of $88.9 billion on newspaper, television, and radio ads. The total amount spent on television and radio ads was only $3.1 billion more than the amount spent on newspaper ads alone. The amount spent on newspaper ads was $5.4 billion more than what was spent on television ads. How much was spent on each form of advertising? (Hint: Let the variables represent numbers of billions of dollars.) How much was spent on newspaper ads? $ nothing billion How much was spent on television ads? $ nothing billion How much was spent on radio ads? $ nothing billion
[tex]\bf \begin{cases} t=\textit{tv ads}\\ r=\textit{radio ads}\\ n=\textit{newspaper ads} \end{cases}~\hspace{7em}t+r+n=88.9 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{tv, radio ads 3.1 more than newspaper}}{t+r=n+3.1}~\hfill \stackrel{\textit{newspaper ads 5.4 more than tv's}}{\boxed{n} = t+5.4}[/tex]
[tex]\bf \stackrel{\textit{substituting on the 2nd equation}}{t+r=\boxed{t+5.4}+3.1}\implies ~~\begin{matrix} t \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~+r=~~\begin{matrix} t \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~+8.5\implies \blacktriangleright r=8.5 \blacktriangleleft \\\\\\ \stackrel{\textit{we know that}}{t+r+n=88.9}\implies t+8.5+n=88.9\implies t+n=80.4[/tex]
[tex]\bf \boxed{n}=80.4-t~\hspace{7em}\stackrel{\textit{substituting on the 3rd equation}}{n=t+5.4\implies \boxed{80.4-t}=t+5.4} \\\\\\ 75-t=t\implies 75=2t\implies \cfrac{75}{2}=t\implies \blacktriangleright 37.5=t \blacktriangleleft \\\\\\ \stackrel{\textit{since we know that}}{n = t+5.4}\implies n=37.5+5.4\implies \blacktriangleright n=42.9 \blacktriangleleft[/tex]
The amounts spent on Newspaper, Television and Radio ads are $40.3 billion, $34.9 billion, and $13.7 billion, respectively.
Explanation:Let us represent the three advertising channels - newspaper, television, and radio - as N, T, and R respectively. Given that total spending on advertising is $88.9 billion, we can write this formally as N + T + R = 88.9 (equation 1). The problem also mentions that total spending on television and radio ads is $3.1 billion more than spending on newspaper ads. This can be written formally as T + R = N + 3.1 (equation 2). Lastly, it is given that spending on newspaper ads is $5.4 billion more than what was spent on television ads. Writing this we get N = T + 5.4 (equation 3). Substituting equation 3 into equation 2, we get (T + 5.4) + R = T + R + 3.1. This simplifies to T - R = 2.3. Solving these equations, we find that $40.3 billion is spent on Newspaper ads, $34.9 billion is spent on Television ads, and $13.7 billion is spent on Radio ads.
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Solve the following ODE's: c) y* - 9y' + 18y = t^2
Answer:
y = [tex]C_1e^{3t}+C_2e^{6t}[/tex] + [tex]\dfrac{1}{18}(t^2+\frac{2t}{6} + \frac{2}{36}+\frac{2t}{3}+\frac{2}{18}+\frac{2}{9})[/tex]
Step-by-step explanation:
y''- 9 y' + 18 y = t²
solution of ordinary differential equation
using characteristics equation
m² - 9 m + 18 = 0
m² - 3 m - 6 m+ 18 = 0
(m-3)(m-6) = 0
m = 3,6
C.F. = [tex]C_1e^{3t}+C_2e^{6t}[/tex]
now calculating P.I.
[tex]P.I. = \frac{t^2}{D^2 - 9D +18}[/tex]
[tex]P.I. = \dfrac{t^2}{(D-3)(D-6)}\\P.I. =\dfrac{1}{18}(1-\frac{D}{3})^{-1}(1-\frac{D}{6})^{-1}(t^2)\\P.I. =\dfrac{1}{18}(1-\frac{D}{3})^{-1}(1+\frac{D}{6}+\frac{D^2}{36}+....)(t^2)\\P.I. =\dfrac{1}{18}(1-\frac{D}{3})^{-1}(t^2+\frac{2t}{6} + \frac{2}{36})\\P.I. =\dfrac{1}{18}(1+\frac{D}{3}+\frac{D^2}{9}+....)(t^2+\frac{2t}{6} + \frac{2}{36})\\P.I. =\dfrac{1}{18}(t^2+\frac{2t}{6} + \frac{2}{36}+\frac{2t}{3}+\frac{2}{18}+\frac{2}{9})[/tex]
hence the complete solution
y = C.F. + P.I.
y = [tex]C_1e^{3t}+C_2e^{6t}[/tex] + [tex]\dfrac{1}{18}(t^2+\frac{2t}{6} + \frac{2}{36}+\frac{2t}{3}+\frac{2}{18}+\frac{2}{9})[/tex]