A statistics professor plans classes so carefully that the lengths of her classes are uniformly distributed between 46.0 and 56.0 minutes. Find the probability that a given class period runs between 50.75 and 51.25 minutes.

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.)

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

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).

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).

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.

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.

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.

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).

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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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

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.

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.

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

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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

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.

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.

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

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]

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.

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.

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

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

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]

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]

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]

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.

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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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:

Using 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°.

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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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.

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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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.

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.

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]

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]