If [tex]H_n[/tex] denotes the [tex]n[/tex]-th hexagonal number, then this is indeed false because [tex]H_2=6[/tex].
Find a compact form for generating function of the sequence 1, 8, 27,.........., k^3,.........
The generating function is [tex]f(x)[/tex] where
[tex]f(x)=\displaystyle\sum_{k=0}^\infty a_kx^k[/tex]
with [tex]a_k=k^3[/tex] for [tex]k\ge0[/tex].
Recall that for [tex]|x|<1[/tex], we have
[tex]g(x)=\dfrac1{1-x}=\displaystyle\sum_{k=0}^\infty x^k[/tex]
Taking the derivative gives
[tex]g'(x)=\dfrac1{(1-x)^2}=\displaystyle\sum_{k=1}^\infty kx^{k-1}=\sum_{k=0}^\infty(k+1)x^k[/tex]
[tex]\implies g'(x)-g(x)=\dfrac x{(1-x)^2}=\displaystyle\sum_{k=0}^\infty kx^k[/tex]
Taking the derivative again, we get
[tex]g''(x)=\dfrac2{(1-x)^3}=\displaystyle\sum_{k=2}^\infty k(k-1)x^{k-2}=\sum_{k=0}^\infty(k^2+3k+2)x^k[/tex]
[tex]\implies g''(x)-3g'(x)+g(x)=\dfrac{x^2+x}{(1-x)^3}=\displaystyle\sum_{k=0}^\infty k^2x^k[/tex]
Take the derivative one last time to get
[tex]g'''(x)=\dfrac6{(1-x)^4}=\displaystyle\sum_{k=3}^\infty k(k-1)(k-2)x^{k-3}=\sum_{k=0}^\infty(k^3+6k^2+11k+6)x^k[/tex]
[tex]\implies g'''(x)-6g''(x)+7g'(x)-g(x)=\dfrac{x^3+4x^2+x}{(1-x)^4}=\displaystyle\sum_{k=0}^\infty k^3x^k[/tex]
So the generating function is
[tex]\boxed{f(x)=\dfrac{x^3+4x^2+x}{(1-x)^4}}[/tex]
Problem Page
Find the slope and the y -intercept of the line.
Write your answers in simplest form.
2x + 5y = -3
Answer:
slope -2/5
y-intercept -3/5
Step-by-step explanation:
Slope-intercept form of a line is y=mx+b where m is the slope and b is the y-intercept.
Our goal is to write 2x+5y=-3 into y=mx+b to determine the slope and y-intercept.
So we need to isolate y.
2x+5y=-3
Subtract 2x on both sides:
5y=-2x-3
Divide both side by 5:
[tex]y=\frac{-2}{5}x-\frac{3}{5}[/tex]
Compare this to y=mx+b.
You should see m is -2/5 and b is -3/5
so the slope is -2/5 and the y-intercept is -3/5
Answer:
The slope is: [tex]-\frac{2}{5}[/tex] or [tex]-0.4[/tex]
The y-intercept is: [tex]-\frac{3}{5}[/tex] or [tex]-0.6[/tex]
Step-by-step explanation:
The equation of the line in Slope-Intercept form is:
[tex]y=mx+b[/tex]
Where "m" is the slope of the line and "b" is the y-intercept.
To write the given equation in this form, we need to solve for "y":
[tex]2x + 5y = -3\\\\5y=-2x-3\\\\y=-\frac{2}{5}x-\frac{3}{5}[/tex]
Therefore, you can identify that the slope of this line is:
[tex]m=-\frac{2}{5}=0.4[/tex]
And the y-intercept is:
[tex]b=-\frac{3}{5}=-0.6[/tex]
Many people think that a national lobby's successful fight against gun control legislation is reflecting the will of a minority of Americans. A random sample of 4000 citizens yielded 2250 who are in favor of gun control legislation. Compute the point estimate for the proportion of citizens who are in favor of gun control legislation. Round to four decimal places.
Answer: [tex]\hat{p}=0.5625[/tex]
Step-by-step explanation:
Given : Sample size : [tex]n=4000[/tex]
The number of people who are in favor of gun control legislation =2250
The proportion of people who are in favor of gun control legislation will be :-
[tex]p_0=\dfrac{2250}{4000}=0.5625[/tex]
We assume that the the given situation is normally distributed.
Then , the point estimate for the proportion [tex]\hat{p}[/tex] of citizens who are in favor of gun control legislation is equals to the sample proportion.
i.e. [tex]\hat{p}=0.5625[/tex]
A company is considering a new manufacturing process. It knows that the rate of savings (in dollars per year) from the process will be about S(t) = 3000(t+2), where t is the number of years the process has been in use. Find the total savings during the first year. Find the total savings during the first 5 years. . The total savings during the first year is S (Simplify your answer.)
Answer:
i). [tex]\$ 7500[/tex]
ii).[tex]\$ 67500[/tex]
Step-by-step explanation:
Given in the question-
Saving rate is s(t)= 3000(t+2)
We know that savings in the 1st year can be calculated as
[tex]\int_{0}^{1}3000(t+2).dt[/tex]
[tex]3000\left [ \frac{t^{2}}{2}+2t \right ]_0^1[/tex]
[tex]3000\left [ \frac{1}{2}+2 \right ][/tex]
= [tex]\$ 7500[/tex]
So savings in the first 5 years can be calculated as
[tex]\int_{0}^{5}3000(t+2).dt[/tex]
[tex]3000\left [ \frac{t^{2}}{2}+2t \right ]_0^5[/tex]
[tex]3000\left [ \frac{25}{2}+5 \right ][/tex]
= [tex]\$ 67500[/tex]
Industry standards suggest that 13 percent of new vehicles require warranty service within the first year. Jones Nissan in Sumter, South Carolina, sold 12 Nissans yesterday. (Round your mean answer to 2 decimal places and the other answers to 4 decimal places.) What is the probability that none of these vehicles requires warranty service
Answer: 0.1880
Step-by-step explanation:
The binomial distribution formula is given by :-
[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex]
where P(x) is the probability of x successes out of n trials, p is the probability of success on a particular trial.
Given : The probability of new vehicles require warranty service within the first year : p =0.13.
Number of trials : n= 12
Now, the required probability will be :
[tex]P(x=0)=^{12}C_0(0.13)^{0}(1-0.13)^{12-0}\\\\=(1)(0.13)^{0}(0.87)^{12}=0.188031682201\approx0.1880[/tex]
Thus, the probability that none of these vehicles requires warranty service = 0.1880
The pieces of a 500 piece puzzle are stored in three containers. 220 pieces are in the first container and 180 pieces are in the second container. What percentage of the pieces in the third container?
Answer:
20 percent
Step-by-step explanation:
Total number of pieces in a puzzle = 500
No. of pieces in first container = 220
No. of pieces in second container = 180
Let no. of pieces in the third container be x.
We get,
[tex]220+180+x=500[/tex]
On adding 220 and 180, we get
[tex]400+x=500[/tex]
On transposing 400 to RHS, we get
[tex]x=500-400=100[/tex]
Percentage of pieces in the third container = (no. of pieces in third container/total no. of pieces in a puzzle) [tex]\times 100[/tex]
[tex]=\frac{100}{500}\times 100=\frac{10000}{500}=20[/tex]
Therefore, percentage of pieces in the third container = 20 percent
Your jewelry business orders $1320 in supplies each month. In each month, you can usually sell an earring/ring set for about $55 each. How many sets do you need to sell each month to break even?
If you found that your total profit for a month was $715, how many sets did you sell?
If you had a month where 7 sets were sold, how much profit did you make?
Answer:
Given,
The jewelry business orders $ 1320 in supplies each month.
That is, the invested amount = $ 1320,
Also, the selling price of each earring/ring set = $ 55,
Part 1 : Let x be the number of set that have been sold each month to break even ( in which revenue and invested amount are equal )
So, Total revenue ( the cost of x sets) = 55x,
⇒ 55x = 1320
Divide both sides by 55,
We get,
x = 24
Hence, 24 sets are needed to sell each month to break even.
Part 2 : Let y be the number of sets in which the profit is $ 715,
Total revenue = 55x
Profit = Total revenue - invested amount
⇒ 55x - 1320 = 715,
⇒ 55x = 715 + 1320
⇒ 55x = 2035
⇒ x = 37
Hence, 37 sets are sold for the total profit of $715.
Part 3 :
Revenue in selling 7 sets = 55 × 7 = $ 385
Profit = $ 385 - $ 1320 = - $ 935
Hence, the profit is - $ 935 after selling 7 sets.
A round silo is 55 feet tall and has a 24 foot radius. How high would a load of 38000 cubic feet of grain fill the silo?
Find the volume of the silo.
The formula is: Volume = PI x r^2 x h
Replace volume with the volume of grain and solve for h:
38000 = 3.14 x 24^2 x h
38000 = 3.14 x 576 x h
38000 = 1808.64 x h
Divide both sides by 1808.64
h = 38000 / 1808.64
h = 21.01
The grain would be 21.01 feet ( round to 21 feet.)
Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 3131 in. by 1717 in. by cutting congruent squares from the corners and folding up the sides. Then find the volume.
Answer:
840.02 square inches ( approx )
Step-by-step explanation:
Suppose x represents the side of each square, cut from the corners of the sheet,
Since, the dimension of the sheet are,
31 in × 17 in,
Thus, the dimension of the rectangular box must are,
(31-2x) in × (17-2x) in × x in
Hence, the volume of the box would be,
V = (31-2x) × (17-2x) × x
[tex]=(31\times 17 +31\times -2x -2x\times 17 -2x\times -2x)x[/tex]
[tex]=(527 -62x-34x+4x^2)x[/tex]
[tex]\implies V=4x^3-96x^2 +527x[/tex]
Differentiating with respect to x,
[tex]\frac{dV}{dx}=12x^2-192x+527[/tex]
Again differentiating with respect to x,
[tex]\frac{d^2V}{dx^2}=24x-192[/tex]
For maxima or minima,
[tex]\frac{dV}{dx}=0[/tex]
[tex]\implies 12x^2-192x+527=0[/tex]
By the quadratic formula,
[tex]x=\frac{192 \pm \sqrt{192^2 -4\times 12\times 527}}{24}[/tex]
[tex]x\approx 8\pm 4.4814[/tex]
[tex]\implies x\approx 12.48\text{ or }x\approx 3.52[/tex]
Since, at x = 12.48, [tex]\frac{d^2V}{dx^2}[/tex] = Positive,
While at x = 3.52, [tex]\frac{d^2V}{dx^2}[/tex] = Negative,
Hence, for x = 3.52 the volume of the rectangle is maximum,
Therefore, the maximum volume would be,
V(3.5) = (31-7.04) × (17-7.04) × 3.52 = 840.018432 ≈ 840.02 square inches
also find the measure of BEF as well
Answer:
∠ABC = 84°
∠BEF = 64°
Step-by-step explanation:
∠ABC is supplementary to the 96° angle shown, so is 180° -96° = 84°.
__
∠ABD, marked as (x+y)°, is a vertical angle with ∠EBC, so has the same measure, 96°. ∠BEF, marked as y°, is a vertical angle with the one marked 2x°.
These relationships can be expressed as two equations:
x + y = 962x = yUsing the second of these equations to substitute for y in the first equation, we have ...
x + 2x = 96
x = 96/3 = 32
y = 2x = 2·32 = 64 . . . . . . substitute the value of x into the second equation
Then ∠BEF = 64°.
An experimental psychologist is interested in whether the color of an animal's surroundings affects learning rate. He tests 16 rats in a box with colorful wallpaper. The average rat (of this strain) can learn to run this type of maze in a box without any special coloring in an average of 25 trials, with a variance of 64, and a normal distribution. The mean number of trials to learn the maze, for the rats tested with the colorful wallpaper, is 11.What is the μM ?• A. 8• B. 11• C. 25• D. 64
Answer: C. 25
Step-by-step explanation:
Given : The average rat (of this strain) can learn to run this type of maze in a box without any special coloring : [tex]\mu=25[/tex]
The mean number of trials to learn the maze, for the rats tested with the colorful wallpaper= [tex]M= 11[/tex]
We know that the sampling distribution D is given by :-
[tex]\mu_D=\mu[/tex]
Similarly the mean of the distribution M in the given situation is given by :_
[tex]\mu_M=\mu=25[/tex]
The mean of the distribution M in the given situation is 25. Then the correct option is C.
What is normal a distribution?It is also called the Gaussian Distribution. It is the most important continuous probability distribution. The curve looks like a bell, so it is also called a bell curve.
An experimental psychologist is interested in whether the color of an animal's surroundings affects the learning rate.
He tests 16 rats in a box with colorful wallpaper.
The average rate (of this strain) can learn to run this type of maze in a box without any special coloring in an average of 25 trials, with a variance of 64, and a normal distribution.
The mean number of trials to learn the maze, for the rats tested with the colorful wallpaper, is 11.
We know that the sampling distribution D is given by
μD = μ
Similarly, the mean of the distribution M in the given situation is given by
μD = μ = 25
More about the normal distribution link is given below.
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let A={2, 4, 6, 8} and B={2, 3, 5, 7, 9} compute n(A)
Answer with explanation:
Given two sets
A={2, 4, 6, 8} and B={2, 3, 5, 7, 9}
⇒n(S)=Cardinality of a set
Means the number of distinct elements in a Set is called it's cardinal number.
→In Set A,total number of distinct elements is 4.
n(A)=4
Q8. the average Ferris wheel rotates at 6.9 miles per hour. What circular distance, in feet dose the average Ferris wheel cover in a 5 minutes ride?
Answer:
303.6 feet
Step-by-step explanation:
Given,
The average Ferris wheel rotates at 6.9 miles per hour.
So, the speed of the wheel = 6.9 miles per hour,
We know that,
Distance = Speed × Time
So, the distance covered by the wheel in 5 minutes ( or 1/12 hours because 1 hour = 60 minutes ) ride = [tex]6.9\times \frac{1}{12}[/tex]
[tex]=\frac{6.9}{12}[/tex]
[tex]=0.575\text{ miles}[/tex]
Since, 1 mile = 5280 feet,
Hence, the distance covered by the wheel in 5 minutes = 0.575 × 528 = 303.6 feet.
The average Ferris wheel covers a circular distance of 3024 feet in a 5-minute ride.
Explanation:To calculate the circular distance covered by the average Ferris wheel in a 5-minute ride, we need to convert the speed from miles per hour to feet per minute. There are 5,280 feet in a mile and 60 minutes in an hour, so we can convert 6.9 miles per hour to feet per minute using the formula:
6.9 miles/hour x 5,280 feet/mile x 1 hour/60 minutes = 604.8 feet/minute
Now that we know the Ferris wheel covers 604.8 feet in 1 minute, we can calculate the circular distance covered in 5 minutes by multiplying the feet per minute by the number of minutes:
604.8 feet/minute x 5 minutes = 3024 feet
Therefore, the average Ferris wheel covers a circular distance of 3024 feet in a 5-minute ride.
It is 76 miles from Waterton to Middleton. It is 87 miles from Middleton to Oak Hill. Driving directly, it is 134 miles from Waterton to Oak Hill. It is 39 miles from Oak Hill to Jackson. If Juan drives from Waterton to Middleton, then from Middleton to Oak Hill, and finally home to Waterton, how many miles does he drive?
Answer: 297 miles
Step-by-step explanation:
The drive from Waterton to Middleton is 76 miles, from Middleton to Oak Hill is 87 miles, and from Oak Hill directly to Waterton it is 134 miles.
Then: [tex]76+87+134=297[/tex]
So Juan drives 297 miles.
On a single roll of a pair of dice, what are the odds against rolling a sum of 3? The odds against rolling a sum of 3 are nothing to nothing.
Answer: Odds against rolling a sum of 3 = 17:1
Step-by-step explanation:
On rolling a pair of dice,
Total number of outcomes = 6 × 6 = 36
Outcomes with a sum of 3:
there is only 2 outcomes whose sum is 3, that is, (1,2) and (2,1)
∴ Favorable outcome = 2
Unfavorable outcome = 34
Odds against refers to the ratio of unfavorable outcomes to the favorable outcomes
so,
odds against rolling a sum of 3 = [tex]\frac{unfavorable\ outcomes}{favorable\ outcomes}[/tex]
= [tex]\frac{34}{2}[/tex]
= 17:1
A participant in a cognitive psychology study is given 50 words to remember and later asked to recall as many of the words as she can. She recalls 17 words. What is the (a) variable, (b) possible values, and (c) score?
Answer:
A participant in a cognitive psychology study is given 50 words to remember and she recalls 17 words.
So, here variable will be the number of words the participant can remember, out of 50.
The possible values can be the whole numbers 0, 1, 2, 3, 4, upto... 50.
And the score is 17. That is the score she remembers out of 50.
10. (Section 4.7) John filled his gas tank and noted that the odometer read 38,320.8 miles. After the next filling, the odometer read 38,735.5 miles. It took 14.5 gal to fill the tank. How many miles per gallon did John get?
Find the number of miles he drove by subtracting the odometer readings:
38735.5 - 38320.8 = 414.7 miles.
Now divide the number of miles driven by the number of gallons:
414.7 / 14.5 = 28.6 miles per gallon.
The equations 8x +4y = 32 and 16x +12y = 72 represent the cost for lunch and dinner for a family eating out on vacation. If x is the number of adults and y is the number of children, how many adults are in the family?
Final answer:
To find the number of adults in the family, we need to solve the system of equations. By multiplying the first equation by 2 and subtracting it from the second equation, we can eliminate x and solve for y. Substituting the value of y back into the first equation, we can solve for x. The number of adults in the family is 3.
Explanation:
To find the number of adults in the family, we need to solve the system of equations:
Equation 1: 8x + 4y = 32
Equation 2: 16x + 12y = 72
We can solve this system by first multiplying Equation 1 by 2 to make the coefficients of x in both equations the same. This gives us:
Equation 1 (multiplied by 2): 16x + 8y = 64
Next, we can subtract Equation 1 (multiplied by 2) from Equation 2 to eliminate x:
Equation 2 - Equation 1 (multiplied by 2): (16x + 12y) - (16x + 8y) = 72 - 64
Simplifying the equation, we get:
4y = 8
Dividing both sides by 4, we find:
y = 2
So, there are 2 children in the family. Substituting this value back into Equation 1, we can solve for x:
8x + 4(2) = 32
8x + 8 = 32
8x = 24
Dividing both sides by 8, we find:
x = 3
Therefore, there are 3 adults in the family.
Assume that women's heights are normally distributed with a mean given by mu equals 62.3 in, and a standard deviation given by sigma equals 2.4 in.(a) If 1 woman is randomly selected, find the probability that her height is less than 63 in.(b) If 47 women are randomly selected, find the probability that they have a mean height less than 63 in.
Answer: a) 0.6141
b) 0.9772
Step-by-step explanation:
Given : Mean : [tex]\mu= 62.3\text{ in}[/tex]
Standard deviation : [tex]\sigma = \text{2.4 in}[/tex]
The formula for z -score :
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
a) Sample size = 1
For x= 63 in. ,
[tex]z=\dfrac{63-62.3}{\dfrac{2.4}{\sqrt{1}}}=0.29[/tex]
The p-value = [tex]P(z<0.29)=[/tex]
[tex]0.6140918\approx0.6141[/tex]
Thus, the probability is approximately = 0.6141
b) Sample size = 47
For x= 63 ,
[tex]z=\dfrac{63-62.3}{\dfrac{2.4}{\sqrt{47}}}\approx2.0[/tex]
The p-value = [tex]P(z<2.0)[/tex]
[tex]=0.9772498\approx0.9772[/tex]
Thus , the probability that they have a mean height less than 63 in =0.9772.
A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n equals 9, p equals 0.8, x less than or equals 3
Answer: 0.0031
Step-by-step explanation:
Binomial distribution formula :-
[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex], where P(x) is the probability of x successes in the n independent trials of the experiment and p is the probability of success.
Given : A binomial probability experiment is conducted with the given parameters.
[tex]n=9,\ p=0.8,\ x\leq3[/tex]
Now, [tex]P(x\leq3)=P(3)+P(2)+P(1)+P(0)[/tex]
[tex]=^9C_3(0.8)^3(1-0.8)^{9-3}+^9C_2(0.8)^2(1-0.8)^{9-2}+^9C_1(0.8)^1(1-0.8)^{9-1}+^9C_0(0.8)^0(1-0.8)^9\\\\=\dfrac{9!}{3!6!}(0.8)^3(0.2)^6+\dfrac{9!}{2!7!}(0.8)^2(0.2)^7+\dfrac{9!}{1!8!}(0.8)(0.2)^8+\dfrac{9!}{0!9!}(0.2)^9=0.003066368\approx0.0031[/tex]
Hence, [tex]P(x\leq3)=0.0031[/tex]
Gold used to make jewerly is often a blend of gold, silver, and copper. Consider three alloys of these metals. The first alloy is 75% gold, 5% silver, and 20% copper. The second alloy is 75% gold, 12.5% silver, and 12.5% copper. The third alloy is 37.5% gold and 62.5% silver. If 100 g of the first alloy costs $2500.40, 100 g of the second alloy costsnbsp $ 2537.75, and 100 g of the third alloy costs $ 1550.00, how much does each metal cost?
Answer:
Gold - $33, Silver - $5, Copper - $0.02
Step-by-step explanation:
Let $x be the price of one gram of gold, $y - price of 1 g of silver and $z - price of 1 g of copper.
1. The first alloy is 75% gold, 5% silver, and 20% copper, so in 100 g there are 75 g of gold, 5 g of silver and 20 g of copper. If 100 g of the first alloy costs $2500.40, then
75x+5y+20z=2500.40
2. The second alloy is 75% gold, 12.5% silver, and 12.5% copper, so in 100 g there are 75 g of gold, 12.5 g of silver and 12.5 g of copper. If 100 g of the first alloy costs $2537.75, then
75x+12.5y+12.5z=2537.75
3. The third alloy is 37.5% gold and 62.5% silver, so in 100 g there are 37.5 g of gold and 62.5 g of silver . If 100 g of the first alloy costs $1550.00, then
37.5x+62.5y=1550.00
Solve the system of three equations:
[tex]\left\{\begin{array}{l}75x+5y+20z=2500.40\\75x+12.5y+12.5z=2537.75\\37.5x+62.5y=1550.00\end{array}\right.[/tex]
Find all determinants
[tex]\Delta=\|\left[\begin{array}{ccc}75&5&20\\75&12.5&12.5\\37.5&62.5&0\end{array}\right] \|=28125\\ \\
\Delta_x=\|\left[\begin{array}{ccc}2500.40&5&20\\2537.75&12.5&12.5\\1550.00&62.5&0\end{array}\right] \|=928125\\ \\
\Delta_y=\|\left[\begin{array}{ccc}75&2500.40&20\\75&2537.75&12.5\\37.5&1550&0\end{array}\right] \|=140625\\ \\
\Delta_z=\|\left[\begin{array}{ccc}75&5&2500.40\\75&12.5&2537.75\\37.5&62.5&1550\end{array}\right] \|=562.5\\ \\[/tex]
So,
[tex]x=\dfrac{\Delta_x}{\Delta}=\dfrac{928125}{28125}=33\\ \\\\y=\dfrac{\Delta_y}{\Delta}=\dfrac{140625}{28125}=5\\ \\\\z=\dfrac{\Delta_z}{\Delta}=\dfrac{562.5}{28125}=0.02\\ \\[/tex]
Consider the system of differential equations dxdt=−4ydydt=−4x. Convert this system to a second order differential equation in y by differentiating the second equation with respect to t and substituting for x from the first equation. Solve the equation you obtained for y as a function of t; hence find x as a function of t. If we also require x(0)=4 and y(0)=5, what are x and y?
The general solution of this differential equation is y(t) = c1 cos(5t) + c2 sin(5t), where c1 and c2 are constants determined by the initial conditions.
Differentiating the second equation with respect to t, we get: d^2y/dt^2 = -5 dx/dt, Substituting dx/dt from the first equation, we get: d^2y/dt^2 = -5(-5y) = 25y.
This is a second order differential equation in y. The general solution of this differential equation is y(t) = c1 cos(5t) + c2 sin(5t), where c1 and c2 are constants determined by the initial conditions.
To find x as a function of t, we can substitute y(t) into the first equation and solve for x: dx/dt = -5y = -5(c1 cos(5t) + c2 sin(5t)) , Integrating both sides with respect to t, we get: x(t) = -c1 sin(5t) + c2 cos(5t) + k
where k is a constant of integration. Using the initial conditions x(0) = 4 and y(0) = 1, we can solve for the constants c1, c2, and k: x(0) = -c1 sin(0) + c2 cos(0) + k = c2 + k = 4, y(0) = c1 cos(0) + c2 sin(0) = c1 = 1
Substituting c1 = 1 and c2 + k = 4 into the equation for x, we get:
x(t) = -sin(5t) + 4
So the solution to the system of differential equations with initial conditions x(0) = 4 and y(0) = 1 is x(t) = -sin(5t) + 4 and y(t) = cos(5t).
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A particular fruit's weights are normally distributed, with a mean of 353 grams and a standard deviation of 6 grams. If you pick one fruit at random, what is the probability that it will weigh between 334 grams and 344 grams?
Answer: 0.0660
Step-by-step explanation:
Given : A particular fruit's weights are normally distributed with
Mean : [tex]\mu=353\text{ grams}[/tex]
Standard deviation : [tex]\sigma=6\text{ grams}[/tex]
The formula to calculate the z-score is given by :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
Let x be the weight of randomly selected fruit.
Then for x = 334 , we have
[tex]z=\dfrac{334-353}{6}=-3.17[/tex]
for x = 344 , we have
[tex]z=\dfrac{344-353}{6}=-1.5[/tex]
The p-value : [tex]P(334<x<353)=P(-3.17<z<-1.5)[/tex]
[tex]P(-1.5)-P(-3.17)=0.0668072-0.000771=0.0660362\approx0.0660[/tex]
Thus, the probability that it will weigh between 334 grams and 344 grams = 0.0660.
[15 points] Compute ffR2(x + 1)y2 dA, R = [ 0, 1] x [0,3), by Riemann sum definition. You must use the Riemann sum definition to receive credit.
Looks like the integral is
[tex]\displaystyle\iint_R2(x+1)y^2\,\mathrm dA[/tex]
where [tex]R=[0,1]\times[0,3][/tex]. (The inclusion of [tex]y=3[/tex] will have no effect on the value of the integral.)
Let's split up [tex]R[/tex] into [tex]mn[/tex] equally-sized rectangular subintervals, and use the bottom-left vertices of each rectangle to approximate the integral. The intervals will be partitioned as
[tex][0,1]=\left[0,\dfrac1m\right]\cup\left[\dfrac1m,\dfrac2m\right]\cup\cdots\cup\left[\dfrac{m-1}m,1\right][/tex]
and
[tex][0,3]=\left[0,\dfrac3n\right]\cup\left[\dfrac3n,\dfrac6n\right]\cup\cdots\cup\left[\dfrac{3(n-1)}n,3\right][/tex]
where the bottom-left vertices of each rectangle are given by the sequence
[tex]v_{i,j}=\left(\dfrac{i-1}n,\dfrac{3(j-1)}n\right)[/tex]
with [tex]1\le i\le m[/tex] and [tex]1\le j\le n[/tex]. Then the Riemann sum is
[tex]\displaystyle\lim_{m\to\infty,n\to\infty}\sum_{i=1}^m\sum_{j=1}^nf(v_{i,j})\frac{1-0}m\frac{3-0}n[/tex]
[tex]\displaystyle=\lim_{m\to\infty,n\to\infty}\frac3{mn}\sum_{i=1}^m\sum_{j=1}^n\frac{18}{mn^2}(j-1)^2(i-1+m)[/tex]
[tex]\displaystyle=\lim_{m\to\infty,n\to\infty}\frac{54}{m^2n^3}\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}j^2(i+m)[/tex]
[tex]\displaystyle=\frac92\lim_{m\to\infty,n\to\infty}\frac{(3m-1)(2n^3-3n^2+n)}{mn^3}[/tex]
[tex]\displaystyle=\frac92\left(\lim_{m\to\infty}\frac{3m-1}m\right)\left(\lim_{n\to\infty}\frac{2n^3-3n^2+n}{n^3}\right)=\boxed{27}[/tex]
If a 2-kg block compresses a spring 800mm from its relaxed state, how much potential energy does the block have due to the spring? Assume the spring constant, k = 25 N/m.
Answer:
8 Joule
Step-by-step explanation:
Mass of block = 2 kg
Displacement = x = 800 mm = 0.8 m
Spring constant = k = 25 N/m
Potential Energy of a spring
Work done = Difference in Potential Energy
Work Done = Δ P.E.
[tex]\Rightarrow \Delta\ P.E.=\frac{1}{2}kx^2[/tex]
⇒P.E. = 0.5×25×0.8²
⇒P.E. = 8 Nm = 8 Joule
Here already the spring constant and displacement is given so the mass will not be used while calculating the potential energy.
What is the value of -4x + 12 if x = -4
Answer:
28
Step-by-step explanation:
substitute x for negative four -4(-4)+12=
solve -4 * -4= 16
add 16 and 12 equals 28
Answer:
28
Step-by-step explanation:
-4x + 12
Let x = -4
-4 (-4) +12
16+12
28
Basing your answer on the appearance of the figures below, identify whether the mathematical expression is true or false.
ΔROB ≅ ΔDFE
True or False
Answer:
The mathematical expression is false
Step-by-step explanation:
* Lets use the figure to answer the question
- There are four triangles in the figure
- Δ ROB and Δ PTA appear congruent because:
# The side RO appears equal the side PT
∴ RO ≅ PT
# The side OB appears equal the side TA
∴ OB ≅ TA
# The side RB appears equal the side PA
∴ RB ≅ PA ⇒ SSS
∴ Δ ROB ≅ Δ PTA
- Δ DEF and Δ YXW appear congruent because:
# The side DE appears equal the side YX
∴ DE ≅ YX
# The side EF appears equal the side XW
∴ EF ≅ XW
# The side DF appears equal the side YW
∴ DF ≅ YW
∴ Δ DEF ≅ Δ YXW ⇒ SSS
- Δ ROB and Δ DEF have different shapes and sizes
∵ Δ ROB not appear congruent to Δ DEF
∴ Δ ROB ≠ Δ DEF
∴ The mathematical expression is false
Answer : The mathematical expression is false.
Step-by-step explanation :
As we are given 4 triangles in which ΔROB & ΔPTA and ΔDEF & ΔYXW are appears congruent.
First we have to show that ΔROB and ΔPTA appear congruent.
Side RO appears equal to Side PT
Side OB appears equal to Side TA
Side RB appears equal to Side PA
∴ ΔROB ≅ ΔPTA (by SSS)
Now we have to show that ΔDEF and ΔYXW appear congruent.
Side DE appears equal to Side YX
Side EF appears equal to Side XW
Side DF appears equal to Side YW
∴ ΔDEF ≅ ΔYXW (by SSS)
According to given expression, ΔROB and ΔDEF have different shapes and sizes.
So, ΔROB not appear congruent to ΔDEF
Therefore, the mathematical expression is false.
as a linear cost function. Find the cost function. Assume that the situation can be expressed Fixed cost is $100 20 items cost 500 to produce. The linear cost function is C(x)= |0
Answer:
Production cost is $20 per item.
Step-by-step explanation:
Fixed cost is $100 and 20 items cost $500 to produce.
[tex]C=100+x*production cost[/tex]
[tex]500=100+20*production cost[/tex]
[tex]400=20*production cost[/tex]
Production cost = $20.
So, [tex]C(x)=20x+100[/tex], where C is total cost and x is the number of items produced.
The linear cost function, based on a given fixed cost and the cost to produce a certain number of items, is found by identifying and adding the fixed and variable costs. In this scenario, the mathematical expression for the total cost function is C(x) = $100 + $20(x).
Explanation:To determine the linear cost function for a production scenario with fixed and variable costs, we use the information provided: the fixed cost is $100, and the cost to produce 20 items is $500. Knowing that the cost function is linear, we can express it as C(x) = Fixed Cost + Variable Cost per Item (x), where C(x) is the total cost function and x is the number of items produced.
Since the fixed cost is given as $100, we have C(x) = $100 + Variable Cost per Item (x). To find the variable cost per item, we calculate the difference in total costs when producing 20 items. This is $500 (total cost to produce 20 items) minus the fixed cost of $100, which equals $400. Since this cost is associated with the production of 20 items, we divide $400 by 20 to find the variable cost per item, which is $20. Thus, our variable cost per item is $20.
Now, we combine the fixed cost with the variable cost per item to get the complete linear cost function: C(x) = $100 + $20(x).
The people at a party tried to form teams with the same number of people on each team, but when they tried to split up into teams of 2, 3, 5, or 7, exactly one person was left without a team. What is the smallest amoutn of people who could have been at the party?
Answer:
211 people
Step-by-step explanation:
1. This is a least common multiple question, otherwise known as an LCM question. We know this because if the one extra person had not shown up to the party, all groups would have been formed evenly. This means the amount of people at the party is 1 more than a number 2, 3, 5, and 7 can go into.
2. We also know this is an LCM question because we are being asked for the smallest amount of people who could have possibly attended the party.
3. From this we know the amount of people at the party must be an odd number. If the number were even, there would have been no left over when groups of 2 were formed.
4. The amount of people at the party must end in a 1. This is because all multiples of 5 always end in a 0 or 5. Because there is on extra person, we must add 1 to all multiples of 5 we check. However, 5 + 1 is 6. This is a problem because 6 is an even number, and as we already established, the amount of people at the party must end in an odd number. So, we now know the smallest amount of people at the party will end in a 1.
5. Because we know the largest teams attempted to be formed with 1 left over is teams of 7 people, we only need to check multiples of 7. It is the largest number, so doing this will save us time.
6. Since we know the amount of people at the party must end in a 1, and we are only checking multiples of 7, we only need to check multiples of 7 that end in a 0. This is because any multiple of 7, 7 will go into evenly without a remainder. So we must add 1 to every multiple we check in order to make a remainder of 1. The only number we can add 1 to in order to get 1 is 0, so we only need to check multiples of 7 that end in 0. The only multiples of 7 that end in a 0 are when 7 is multiplied by a ten, ex: 10, 20, 30, 40, 50, ect.
7. Only searching for odd numbers, numbers that end in 1, and multiples of 7 means we only have to check if any possible answer when divided by 3 has a remainder of 1. We only have to check by the number 3 because any number ending in 1 will automatically have a remainder of one for 2 and 5, and because we are using multiple of 7 we don't need to check through 7.
8. Now that we know all our rules, all we need to do is list multiples of 7 that end in 0. Then we will add 1 to them and check to see if they have a remainder of 1 when divided by 3.
Using these rules will narrow our search drastically.
Applicable multiples of 7
(7 × 10) = 70 → 70 + 1 = 71 → 71/3 = 23 R2 → not possible
(7 × 20) = 140 → 140 + 1 = 141 → 141/3 = 47 → not possible
(7 × 30) = 210 → 210 +1 = 211 → 211/3 = 70 R1 → possible
The smallest possible amount of people at the party is 211.
Final answer:
The smallest amount of people who could have been at the party is 210.
Explanation:
To find the smallest amount of people who could have been at the party, we need to find the least common multiple (LCM) of the numbers 2, 3, 5, and 7. The LCM is the smallest number that is divisible by all of the given numbers without leaving a remainder.
Prime factorize each of the given numbers:Therefore, the smallest amount of people who could have been at the party is 210.
When a pair of dice is rolled, what is the probability that the sum of the dice is 5, given that the outcome is not 6? The probability that the sum of the dice is 5 given that the outcome is not 6 is (Type an integer or a simplified fraction.)
The probability that the sum of the two dice rolls is 5 given that the sum is not 6, is calculated by finding the ratio of favorable outcomes to total outcomes, in this case, 4/31.
Explanation:
The subject of this question is probability which comes under Mathematics. This is a high school-level problem. To answer the question, we first need to understand the rules of a die. A die is a cube, and each of its six faces shows a different number of dots from 1 to 6. When the die is thrown, any number from 1 to 6 can turn up. In this case, two dice are being rolled.
When two dice are rolled, the total possible outcomes are 36 (as each die has 6 faces & we have 2 dice, so 6*6=36 possible outcomes). The combinations that yield a sum of 5 are (1,4), (2,3), (3,2), (4,1), so there are 4 such combinations. Now, the outcome is given to be not 6, which means we exclude combinations where the sum is 6. The combinations of 6 are (1,5), (2,4), (3,3), (4,2), and (5,1) -- 5 combinations.
Excluding these combinations, we have 36 - 5 = 31 possible outcomes. So probability that the sum of the dice is 5 given that the outcome is not 6, is favorable outcomes/total outcomes = 4/31.
Learn more about Probability here:https://brainly.com/question/32117953
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