Answer: a) 8359 b) 384
Step-by-step explanation:
Given : Significance level : [tex]\alpha=1-0.95=0.05[/tex]
Critical value : [tex]z_{\alpha/2}}=\pm1.96[/tex]
Margin of error : [tex]E=0.01[/tex]
a) If previous estimate of proportion : [tex]p=0.32[/tex]
Formula to calculate the sample size needed for interval estimate of population proportion :-
[tex]n=p(1-p)(\frac{z_{\alpha/2}}{E})^2[/tex]
[tex]\Rightarrow\ n=0.32(1-0.32)(\frac{1.96}{0.01})^2=8359.3216\approx 8359[/tex]
Hence, the required sample size would be 8359 .
b) If she does not use any prior estimate , then the formula to calculate sample size will be :-
[tex]n=0.25\times(\frac{z_{\alpha/2}}{E})^2\\\\\Rightarrow\ n=0.25\times(\frac{1.96}{0.05})^2=384.16\approx384[/tex]
Hence, the required sample size would be 384 .
find the value of solid cylinder figure not including cut out: cylinder is 9.7 ft in height, 7 feet in base length, with 4 ft hole cut out.
Answer:
251.28 cubic feet
Step-by-step explanation:
The height of the cylinder is 9.7 ft.
The base length is 7 feet. So, the radius(R) = [tex]\frac{7}{2} = 3.5 feet[/tex]
The length of 4 feet cylinder cut out. So, the radius of the cut cylinder (r) = [tex]\frac{4}{2} = 2 feet[/tex]
We have to find the volume of solid cylinder figure without cutting part.
= Volume of the whole cylinder - Volume of the hole cut
We know that volume of a cylinder is [tex]\pi *r^2*h[/tex]
Using this formula,
= [tex]\pi *R^2*h - \pi *r^2*h[/tex]
= [tex]\pi h [R^2 - r^2][/tex]
Here π = 3.14, R = 3.5, r = 2 and h = 9.7
Plug in these values in the above, we get
= [tex]3.14*9.7 [3.5^2 - 2^2]\\= 30.458[12.25 - 4]\\= 30.458[8.25]\\= 251.2785 ft^3[/tex]
When round of to the nearest hundredths place, we get
So, the volume of solid cylinder figure not including cut out= 251.28 cubic feet
On a recent road trip, it took John 3.53.5 h to drive from a suburb of Phoenix to a small town outside of Yuma. According to his dashboard computer, he averaged 8585 mph on the drive. John's friend in Yuma is European, and he asked for the distance that John drove in kilometers. How far did John drive in kilometers?
Answer:
He drove approximately 478.78 km
Step-by-step explanation:
We know that,
Distance = Speed × time,
Given,
Time taken by john in driving = 3.5 hours,
His average speed = 85 mph,
So, the total distance he drove = 3.5 × 85 = 297.5 miles,
Since, 1 miles = 1.60934 km,
Thus, the total distance he drove = 1.60934 × 297.5 = 478.77865 km ≈ 478.78 km
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]
The student council at a college is made up of four freshmen, five sophomores, six juniors, and seven seniors. A yearbook photographer would like to line up two council members from each class for a picture. How many different pictures are possible if each group of classmates stands together?
Answer:
7257600
Step-by-step explanation:
Number of freshmen in the student council= 4
Number of sophomores in the student council= 5
Number of juniors in the student council= 6
Number of seniors in the student council= 7
Ways of choosing council members
⁴C₂×⁵C₂×⁶C₂×⁷C₂
[tex]^4C_2=\frac{4!}{(4-2)!2!}\\=\frac{24}{4}=6\\\\^5C_2=\frac{5!}{(5-2)!2!}\\=\frac{120}{12}=10\\\\^6C_2=\frac{6!}{(6-2)!2!}\\=\frac{720}{48}=15\\\\^7C_2=\frac{7!}{(7-2)!2!}\\=\frac{5040}{240}=21[/tex]
⁴C₂×⁵C₂×⁶C₂×⁷C₂=6×10×15×21=18900
Ways of lining up the four classes=4!=1×2×3×4=24
Ways of lining up members of each class=2⁴=2×2×2×2=16
Pictures are possible if each group of classmates stands together
⁴C₂×⁵C₂×⁶C₂×⁷C₂×4!×2⁴
=18900×24×16
=7257600
There are 453600 possible different pictures
How to determine the number of different pictures?
The given parameters are:
Freshmen = 4
Sophomores = 5
Juniors = 6
Seniors = 7
Two council members are to be selected from each group.
So, the number of ways this can be done is:
n = ⁴C₂×⁵C₂×⁶C₂×⁷C₂
Apply combination formula, and evaluate the product
n = 18900
Each group are to stand together.
There are 4! ways to arrange the 4 groups.
So, the total number of pictures is:
Total = 4! * 18900
Evaluate the product
Total = 453600
Hence, there are 453600 possible different pictures
Read more about combination at:
https://brainly.com/question/11732255
A real estate firm owns the Kasumba Garden Apartments which consists of 90 apartments. At $350 per month every apartment can be rented. However, for each $10 per month increase there will be two vacancies with no possibility of filling them. The firm wants to receive $31 980 per month from rents. What rent should be charged for each apartment?
Answer:
The rent should be $ 390 or $ 410.
Step-by-step explanation:
Given,
The original monthly rent of an apartment = $350,
Also, the original number of apartment that could be filled = 90,
Let the rent is increased by x times of $ 10,
That is, the new monthly rent of an apartment =( 350 + 10x ) dollars
Since, for each $10 per month increase there will be two vacancies with no possibility of filling them.
Thus, the new number of filled apartments = 90 - 2x,
Hence, the total revenue of the firm = ( 90 - 2x )(350 + 10x ) dollars,
According to the question,
( 90 - 2x )(350 + 10x ) = 31,980
[tex]90(350)+90(10x)-2x(350)-2x(10x)=31980[/tex]
[tex]31500+900x-700x-20x^2=31980[/tex]
[tex]-20x^2+200x+31500-31980=0[/tex]
[tex]20x^2-200x+480=0[/tex]
By the quadratic formula,
[tex]x=\frac{200\pm \sqrt{(-200)^2-4\times 20\times 480}}{40}[/tex]
[tex]x=\frac{200\pm \sqrt{1600}}{40}[/tex]
[tex]x=\frac{200\pm 40}{40}[/tex]
[tex]\implies x=\frac{200+40}{40}\text{ or }x=\frac{200-40}{40}[/tex]
[tex]\implies x=6\text{ or } x =4[/tex]
Hence, the new rent of each apartment, if x = 6, is $ 410,
While, if x = 4, is $ 390
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]
Find x.
A.4
B.18
C.12
D.38
The inside angle is half the outside angle.
2x +2 = 76 /2
2x +2 = 38
Subtract 2 from each side:
2x = 36
Divide both sides by 2:
x = 36 /2
x = 18
The answer is B.
Provide a counterexample:
If a and b are integers where a is divisible by b and b is divisible by a, then a = b.
Answer:
[tex]a=2 \quad \text{and} \quad b=-2[/tex]
Step-by-step explanation:
Take [tex]a=2 \quad \text{and} \quad b=-2[/tex], note that
[tex]2=(-1)\cdot(-2)[/tex]
hence b divides a. On the other hand, we have that
[tex]-2=(-1)\cdot2[/tex]
which tells us that a divides b. Moreover, [tex]a=2 \neq -2=b[/tex].
Write an equation of the circle with center (4,5) and radius 9.
Answer:
I'm going to write both of these because maybe you have a fill in the blank question. I don't know.
[tex](x-4)^2+(y-5)^2=9^2[/tex]
Simplify:
[tex](x-4)^2+(y-5)^2=81[/tex]
Step-by-step explanation:
[tex](x-h)^2+(y-k)^2=r^2[/tex] is the equation of a circle with center (h,k) and radius r.
You are given (h,k)=(4,5) because that is the center.
You are given r=9 because it says radius 9.
Let's plug this in.
[tex](x-4)^2+(y-5)^2=9^2[/tex]
Simplify:
[tex](x-4)^2+(y-5)^2=81[/tex]
Answer:
(x-4)^2 + (y-5)^2 = 9^2
or
(x-4)^2 + (y-5)^2 =81
Step-by-step explanation:
The equation for a circle is (x-h)^2 + (y-k)^2 = r^2
Where (h,k) is the center and r is the radius
(x-4)^2 + (y-5)^2 = 9^2
or
(x-4)^2 + (y-5)^2 =81
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
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.
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.
2. A random sample of 30 households was selected as part of a study on electricity usage, and the number of kilowatt-hours (kWh) was recorded for each household in the sample for the March quarter of 2017. The average usage was found to be 375kWh. From past years in the March quarter the population standard deviation of the usage was 81k Assuming the standard deviation is unchanged and that the usage is normally distributed ; A. Determine the interval of 95% confidence for the average kilowatt-hours for the population. B. Determine the 99% confidence interval. C. With a confidence level of 90%, what would the minimum sample size need to be in order for the true mean of the heights to be less than 20kWh from the sample mean? (This would mean an error amount of 20kWh.)
Answer:
Step-by-step explanation:
Given that n =30, x bar = 375 and sigma = 81
Normal distribution is assumed and population std dev is known
Hence z critical values can be used.
For 95% Z critical=1.96
Margin of error = [tex]1.96(\frac{81}{\sqrt{30} } )=29[/tex]
Confidence interval = 375±29
=(346,404)
B) 99% confidence
Margin of error = 2.59*Std error =38
Confidence interval = 375±38
=(337, 413)
C) For 90%
Margin of error = 20
Std error = 20/1.645 = 12.158
Sample size
[tex]n=(\frac{81}{12.158} )^2\\=44.38[/tex]
Atleast 44 people should be sample size.
To determine the confidence intervals for the average kilowatt-hours, a formula is used that includes the sample mean, Z-values, population standard deviation, and sample size. For a 95% confidence level, the interval is 324.95 to 425.05 kWh, and for a 99% confidence level, the interval is 311.01 to 438.99 kWh. To have a confidence level of 90% with a maximum error amount of 20kWh, the minimum sample size required is approximately 35 households.
Explanation:A. Determine the interval of 95% confidence for the average kilowatt-hours for the population:
To determine the interval of 95% confidence, we can use the formula:
95% confidence interval = sample mean ± (Z-value) * (population standard deviation / √sample size)
Substituting the given values, we have:
95% confidence interval = 375 ± (1.96) * (81 / √30) = 324.95 to 425.05 kWh
B. Determine the 99% confidence interval:
Using the same formula, but with a Z-value of 2.57 (corresponding to 99% confidence), we have:
99% confidence interval = 375 ± (2.57) * (81 / √30) = 311.01 to 438.99 kWh
C. Minimum sample size for a confidence level of 90% and an error amount of 20kWh:
To determine the minimum sample size, we can rearrange the formula for the confidence interval and solve for the sample size:
Sample size = ((Z-value) * (population standard deviation / error amount))^2
Substituting the given values, we have:
Sample size = ((1.645) * (81 / 20))^2 = 34.64 or approximately 35 households
Learn more about Confidence intervals here:https://brainly.com/question/34700241
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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]
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
What is the median for the set of data? Ages Stem Leaves
1 0, 3, 6
2 0, 1, 3, 7, 7, 8, 9
3 0, 2, 3, 3, 3, 3, 8, 9
4 6, 6, 6, 8 1|0 = 10 years old
30
31
32
33
Answer:
This is a stem leaf data, in which the stem generally stands for the "tens" place value while the leaf stands for the "ones" place value.
Expand the data, and find the median by finding the middle number:
10, 13, 16, 20, 21, 23, 27, 27, 28, 29, 30, 32, 33, 33, 33, 33, 38, 39, 46, 46, 46, 48
There are 22 numbers in all. To find the Median when there is a even amount of numbers, Find the two middle numbers, and find the mean of the two numbers:
(32 + 30)/2 = (62)/2 = 31
31 is your answer.
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a slice of bread has 1.5 g of fat, 19 g carbohydrate and 4 g protein. how many kcal in the slice? please show all of you work in a dimensional analysis format.
Answer:
The answer is 105.5 kcal/slice of bread.
Step-by-step explanation:
The kcal per gram of:
protein = 4kcal/g.carbohydrate = 4kcal/g.fat = 9kcal/g.Now, you have:
1.5 g of fat + 19g of carbohydrate + 4g of protein = 1 slice of bread.If we multiply the amount of components by his kcal:
1.5g * 9kcal/g + 19g * 4kcal/g + 4g * 4kcal/g;Now gram/gram = 1, and we can cancel the grams in the equation:
1.5* 9kcal + 19 * 4kcal + 4 * 4kcal = 105.5kcal;Finally, the result of the kcal of the slice bread is:
105.5 kcal/slice of bread.U = {a, b, c, d, e, f, g} A = {a, c, e, g} B = {a, b, c, d} Find A ∩ B
[tex]A\cap B=\{x:x\in A \wedge x\in B\}[/tex]
[tex]\large\boxed{A\cap B=\{a,c\}}[/tex]
A bit out of practice, but how to go about finding the origin and vertix of this equation
9x^(2)+16y^(2)=144
Answer:Explained Below
Step-by-step explanation:
The given equation is similar to an ellipse which is in the form of
[tex]\frac{x^2}{a^2}[/tex]+[tex]\frac{y^2}{b^2}[/tex]=1
where
2a=length of major axis
2b=length of minor axis
Here after rearranging the given equation we get
[tex]\frac{x^2}{\frac{144}{9}}[/tex]+[tex]\frac{y^2}{\frac{144}{16}}[/tex]=1
[tex]\frac{x^2}{16}[/tex]+[tex]\frac{y^2}{9}[/tex]=1
[tex]\frac{x^2}{4^2}[/tex]+[tex]\frac{y^2}{3^2}[/tex]=1
therefore its origin is (0,0)
and vertices are[tex]\left ( \pm4,0\right )[/tex]&[tex]\left ( 0,\pm3\right )[/tex]
We can find origin by checking what is with x in the term [tex]\left ( x-something\right )^{2}[/tex]
same goes for y
for [tex]\left ( x-2\right )^{2}[/tex] here 2 is the x coordinate of ellipse
and for vertices Each endpoint of the major axis is vertices and each endpoint of minor axis is co-vertices
[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 condition on the probability of success p that will guarantee the histogram of a binomial distribution is symmetrical about x=n/2. Justify your answer.
Answer:
The probability of success is 1/2.
Step-by-step explanation:
The histogram of a binomial distribution has a mode of n×p. For that to be n/2, the value of p must be 1/2.
Use Stokes' Theorem to evaluate S curl F · dS. F(x, y, z) = 5y cos(z) i + ex sin(z) j + xey k, S is the hemisphere x2 + y2 + z2 = 4, z ≥ 0, oriented upward. Step 1 Stokes' Theorem tells us that if C is the boundary curve of a surface S, then curl F · dS S = C F · dr Since S is the hemisphere x2 + y2 + z2 = 4, z ≥ 0 oriented upward, then the boundary curve C is the circle in the xy-plane, x2 + y2 = 4 Correct: Your answer is correct. seenKey 4 , z = 0, oriented in the counterclockwise direction when viewed from above. A vector equation of C is r(t) = 2 Correct: Your answer is correct. seenKey 2 cos(t) i + 2 Correct: Your answer is correct. seenKey 2 sin(t) j + 0k with 0 ≤ t ≤ 2π.
By Stokes' theorem, the integral of the curl of [tex]\vec F[/tex] across [tex]S[/tex] is equal to the integral of [tex]\vec F[/tex] along the boundary of [tex]S[/tex], call it [tex]C[/tex]. Parameterize [tex]C[/tex] by
[tex]\vec r(t)=2\cos t\,\vec\imath+2\sin t\,\vec\jmath[/tex]
with [tex]0\le t\le2\pi[/tex]. So we have
[tex]\displaystyle\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S=\int_C\vec F\cdot\mathrm d\vec r[/tex]
[tex]=\displaystyle\int_0^{2\pi}(10\sin t\cos 0\,\vec\imath+e^{2\cos t}\sin0\,\vec\jmath+2\cos t\,e^{2\sin t}\,\vec k)\cdot(-2\sin t\,\vec\imath+2\cos t\,\vec\jmath)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^{2\pi}-20\sin^2t\,\mathrm dt[/tex]
[tex]=\displaystyle-10\int_0^{2\pi}(1-\cos2t)\,\mathrm dt=\boxed{-20\pi}[/tex]
The problem makes use of Stokes' theorem to evaluate a given field over a hemisphere. We established the boundary curve of the surface and described it using a vector equation.
Explanation:This problem can be solved using Stokes' theorem which asserts that the magnetic field flux through a surface is related to the circulation of the field encircling that surface. Stokes' theorem can be written in this form ∫ S curl F · dS = ∫ C F · dr. Given the field F(x, y, z) = 5y cos(z) i + ex sin(z) j + xey k and the hemisphere S defined by x² + y² + z² = 4, z ≥ 0, we need to look for its boundary curve C. C here is the circle in the xy-plane defined by x² + y² = 4, z = 0. We can describe this boundary using a vector equation r(t) = 2 cos(t) i + 2 sin(t) j + 0k with 0 ≤ t ≤ 2π.
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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
you flip a spinner that has 4 differently colored regions (red, white, blue, green) all equal in area, and toss 1 die and count the dots on top (1 thourgh 6 possible). How many color-dot outcomes are possible?
Answer:
24 ways
Step-by-step explanation:
Two different events
1) flip a spinner with 4 different colors regions.
2) tossing a die with 6 outcomes
to calculate number of colored dots possible
note here both action are independent of each other
by the principal of counting we can say
if an act is performed in m ways and another act can be performed in n ways the both the act simultaneously can be performed in [tex]m\times n[/tex] ways.
here act 1 has m=4 ways and act n= 6 ways
hence number of ways of getting colored dots = [tex]4\times6[/tex] ways
= 24 ways
The total number of possible outcomes when flipping a spinner with 4 differently colored regions and tossing a die is 24, calculated by multiplying the number of possible outcomes from the spinner (4) and the die (6).
Explanation:The subject of the question is the calculation of possible outcomes in a probability scenario involving a spinner and a die. A spinner with 4 differently colored areas can give 4 outcomes (red, white, blue, green), and tossing a die can result in 6 outcomes (1, 2, 3, 4, 5, 6).
To find the total number of possible outcomes, we simply multiply the number of possible outcomes from the spinner and the die: 4 (from the spinner) times 6 (from the die).
So, there are 24 color-dot outcomes possible when flipping a spinner with 4 differently colored areas and tossing a die.
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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).
Why do you think students have difficulty understanding the concept of a mathematical function?
Answer:
Because they have never had to express one quantity in terms of another. The idea of such a relationship is completely new, as is the vocabulary for expressing such relationships.
Step-by-step explanation:
"Function" is a simple concept that says you can relate two quantities, and you can express that relationship in a number of ways. (ordered pairs, table, graph)
The closest experience most students have with functions is purchasing things at a restaurant or store, where the amount paid is a function of the various quantities ordered and the tax. Most students have never added or checked a bill by hand, so the final price is "magic", determined solely by the electronic cash register. The relationship between item prices and final price is completely lost. Hence the one really great opportunity to consider functions is lost.
Students rarely play board games or counting games (Monopoly, jump rope, jacks, hide&seek) that would give familiarity with number relationships. They likely have little or no experience with the business of running a lemonade stand or making and selling items. Without these experiences, they are at a significant disadvantage when it comes to applying math to their world.
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.)
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]
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.
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