A student standing on the edge of a cliff throws a rock downward at a speed of 7.5 m/s at an angle 40° below the horizontal. It takes the rock 2.4 seconds to hit the ground. How tall is the cliff?

Answer: u (sub d) is inequal to zero

Step-by-step explanation: Because this is a paired t-test, our alternative hypothesis would be u(sub d) is inequal to zero.

Final answer:

The alternative hypothesis for a study on the effectiveness of a diet program would express the expectation of a statistically significant change in fitness level scores, likely a decrease if lower scores indicate better fitness, after participating in the program compared to before.

Explanation:

The correct alternative hypothesis for a study that aims to measure the effectiveness of a diet program in managing weight would look at whether there is a statistically significant difference in the fitness level scores of the participants before and after following the program. As the question suggests evaluating the effectiveness of a diet program, we are particularly interested in seeing an improvement, which would mean expecting a lower score after the program if the score represents a measure where lower is better.

Thus, the alternative hypothesis (H1) should reflect the expectation of improvement. If the fitness score is such that a lower score indicates better fitness, the alternative hypothesis would be:

H1: The mean fitness level score after the program is lower than the mean fitness level score before the program.

This implies that the diet program is effective in improving fitness levels. On the contrary, if a higher fitness score indicates better fitness, the alternative hypothesis would be framed to reflect an expected increase in the score after the program.

Answer: The explanation is as follows:

Step-by-step explanation:

(a) [8: 9, 3, 2, 1]

q = 8

Here, coalition is as follows:

[P1, P2, P3, P4] = [9, 3, 2, 1]

for the above coalition, the combined weight is

[P1, P2, P3, P4] = 9+3+2+1 = 15 ⇒ combined weight

For simplified weighted voting system;

q = combined weight ⇒ both the terms have to be equal for a simplified weighted voting system.

But, here 8 ≠ 15

∴ It is not a simplified weighted voting system.

(b) [20: 8, 6, 3, 2, 1]

q = 20

Here, coalition is as follows:

[P1, P2, P3, P4, P5] = [8, 6, 3, 2, 1]

for the above coalition, the combined weight is

[P1, P2, P3, P4, P5] = 8+6+3+2+1 = 20 ⇒ combined weight

For simplified weighted voting system;

q = combined weight

Since,  20 = 20

∴ It is a simplified weighted voting system.

Answer:

The probability[tex]0.75095[/tex] and the parameter [tex]p=0.629[/tex]

Step-by-step explanation:

The formula for probability  in a binomial distribution is:

[tex]b(x;n,p)=\frac{n!}{x!(n-x)!}\ast p^{x}\ast(1-p)^{n-x}[/tex]

where p is the probability of success (ticket with popcorn coupon), n is the number of trials (tickets bought) and x the number of successes desired. In this case p=0.629 (probability of buying a movie ticket with coupon), n=29,  and x=17,18,19, ...29.

[tex]b(17;29,0.629)=\frac{29!}{17!(29-17)!}\ast0.629^{17}\ast(1-0.629)^{29-17}=0.133\,25\\ b(18;29,0.629)=\frac{29!}{18!(29-18)!}\ast0.629^{18}\ast(1-0.629)^{29-18}=0.150\,61[/tex]

[tex]b(19;29,0.629)=\frac{29!}{19!(29-19)!}\ast0.629^{19}\ast(1-0.629)^{29-19}=0.147\,84\\ b(20;29,0.629)=\frac{29!}{20!(29-20)!}\ast0.629^{20}\ast(1-0.629)^{29-20}=0.125\,32 \\ b(21;29,0.629)=\frac{29!}{21!(29-21)!}\ast0.629^{21}\ast(1-0.629)^{29-21}=0.091\,06 \\ b(22;29,0.629)=\frac{29!}{22!(29-22)!}\ast0.629^{22}\ast(1-0.629)^{29-22}=0.056\,14[/tex]

[tex]b(23;29,0.629)=\frac{29!}{23!(29-23)!}\ast0.629^{23}\ast(1-0.629)^{29-23}=2.896\,8\times10^{-2} \\ b(24;29,0.629)=\frac{29!}{24!(29-24)!}\ast0.629^{24}\ast(1-0.629)^{29-24}=1.227\,8\times10^{-2}\\ b(25;29,0.629)=\frac{29!}{25!(29-25)!}\ast0.629^{25}\ast(1-0.629)^{29-25}=4.163\,4\times10^{-3} \\ b(26;29,0.629)=\frac{29!}{26!(29-26)!}\ast0.629^{26}\ast(1-0.629)^{29-26}=1.085\,9\times10^{-3} \\ b(27;29,0.629)=\frac{29!}{27!(29-27)!}\ast0.629^{27}\ast(1-0.629)^{29-27}=2.045\,7\times10^{-4}[/tex]

[tex]b(28;29,0.629)=\frac{29!}{28!(29-28)!}\ast0.629^{28}\ast(1-0.629)^{29-28}=2.477\,4\times10^{-5} \\ b(29;29,0.629)=\frac{29!}{29!(29-29)!}\ast0.629^{29}\ast(1-0.629)^{29-29}=1.448\,3\times10^{-6}[/tex]

The probability of more than 16 is equal to the sum of the probability of x=17, 17,18,19, ...29.

[tex]b(x>16;29,0.629)=0.13325+0.15061+0.14784+0.12532+0.09106+0.05614+2.8968\times10^{-2}+1.2278\times10^{-2}+4.1634\times10^{-3}+1.0859\times10^{-3}+2.0457\times10^{-4}+2.4774\times10^{-5}+1.4483\times10^{-6}=0.75095[/tex]

The numerical value of the parameter p in this binocular distribution scenario, where getting a popcorn coupon is defined as a success, is 0.629.

In this binomial distribution scenario, we are considering buying a movie ticket with a popcorn coupon as a success. The probability of success (p) is given as 0.629. Therefore, in this context, the numerical value of the parameter p for the binomial distribution is 0.629.

This means that each time a ticket is bought, there is a 0.629 chance (or 62.9%) of getting a popcorn coupon with the ticket. This probability stays constant with each new ticket purchase. In other words, the purchase of one ticket does not influence the likelihood of the outcome of the next ticket. This makes the scenario suitable to be modeled using a binomial distribution.

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Check the picture below.

Answer: SECOND OPTION.

Step-by-step explanation:

It is important to remember that a tangent to a circle is a line that touches it at one point. This point is called "Point of tangency". By definition, the angle between the tangent and the radius is 90 degrees.

In this case you can observe that EF is the radius of this circle, therefore, the angle between EF and FH measures 90 degrees.

Based on this, you can say the following:

[tex]\angle EFH=90\°[/tex]

This matches with the second option.

Answer:

my money= $192.5

Natalie's money= $110

Step-by-step explanation:

let my and Natalie's money be 7x and 4x respectively.

Now i gave $15 dollar to Natalie so now

my and Natalie's money will be (7x-15) and (4x+15) respectively.

now my money is 20% more than Natalie

therefore,

[tex]\frac{7x-15}{4x+15} =\frac{6}{5}[/tex]

now calculating for x  we get

x= 27.5

since, my and Natalie's money be 7x and 4x respectively

putting x=27.5

my money= $192.5

Natalie's money= $110

Answer: 0.04395

Explanation:

Given: 10 true-false questions.

So, we will have 50% chances (probability = 0.5) of being correct.

Prob( Exactly 80% score) = Prob (exactly 8 answers correct)

As we observe, if X= number of correct answers, then X~ Binomial (n=10, p=0.5)

So, Prob( Exactly 80% score) = Prob (exactly 8 answers correct)

=[tex]\binom{10}{8}\times(1/2)^{8}}\times(1/2)^{2}[/tex]

= 0.0439453125

= 0.04395

To find the point on the plane that is nearest to (2,0,1), we minimize the squared distance between the two points using partial derivatives and set them equal to 0. The values of x, y, and z for the point are x = 28/13, y = 3/26, and z = 27/26.

To find the point on the plane that is nearest to (2,0,1), we need to find the coordinates that satisfy the equation 4x+3y+z=10 and minimize the distance between the point and (2,0,1).

This can be done by minimizing the squared distance between the two points. Using the formula for distance, we get the squared distance as:

d^2 = (x-2)^2 + y^2 + (z-1)^2

To minimize the squared distance, we can find the partial derivatives with respect to x, y, and z and set them equal to 0.

Solving these equations, we find that x = 28/13, y = 3/26, and z = 27/26.

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Answer:

Domain: amount of fuel in the airplane's tank (in gallons)

The set of all real numbers from 0 to 200

Range: weight of airplane (In  pounds)

The set of all real numbers from 3000 to 4400

Step-by-step explanation:

We have the following function

[tex]W=7F+3000[/tex]

Where W represents the weight of the plane in pounds and F represents the amount of fuel in gallons.

The domain of a function is the set of values ​​"F" that can be entered in a function W(F) to obtain an output value of W.

In this case the range of the function W(F) is the whole set of values [tex]W_1, W_2, W_3, ..., W_n[/tex] that are obtained for [tex]F_1, F_2, F_3, ..., F_n[/tex]

Note that, in this case, equation W(F) is used to obtain the weight of the airplane from the amount of fuel F.

Then the domain of the function is the amount of fuel in the airplane tank (in gallons). Since the tank can only hold up to 200 gallons, and there are no negative volume units, then the domain is all real numbers between 0 and 200.

The range of the function is the weight of the plane (in pounds). Note that the minimum weight of the airplane with 0 gallons of fuel is 3000 pounds and the maximum weight with the full tank is 4400 pounds.

Then the range is all real numbers between 3000 and 4400

Answer:

[tex](x+1)^{2} +(y-4)^{2} =25[/tex]

Step-by-step explanation:

In order to find the equation of the circle, first we need to know the circle's general equation, which is:

[tex](x-h)^{2} +(y-k)^{2} =r^{2}[/tex] where:

(h,k) is the center of the circle and r the radius of the circle.

Because the problem has given the center (-1,4) then h=-1 and k=4.

We need to find now the radius:

Using the distance equation: [tex]distance=\sqrt{(x2-x1)^{2}+(y2-y1)^{2}}[/tex] and because we have the center coordinates and an extra point (3,7) we can find the radius as:

[tex]distance=\sqrt{(3-(-1))^{2}+(7-4)^{2}}[/tex]

[tex]distance=\sqrt{4^{2}+3^{2}}[/tex]

[tex]distance=\sqrt{16+9}[/tex]

[tex]distance=\sqrt{25}[/tex]

[tex]distance=5[/tex] which means r=5

In conclusion, the equation for the given circle is [tex](x+1)^{2} +(y-4)^{2} =5^{2}[/tex] which also, can be written as [tex](x+1)^{2} +(y-4)^{2} =25[/tex]

Answer:

Option D. $850.64; $91.14

Step-by-step explanation:

Joyce Meadow pays her three workers per week  $160, $470 and $800 respectively.

A year has 52 weeks, Therefore quarter has [tex]\frac{52}{4}[/tex] = 13 weeks.

So for a quarter she pays = 160 × 13 = $2,080, 470 × 13 = $6,110, 800 × 13 = $10,400

State rate is 5.6%

Federal rate is 0.6%

Base is $7,000

It means the unemployment needs to be paid on the first $7000 only

So the state unemployment = (0.056 × 2,080) + (0.056 × 6,110) + (0.056 × 7,000)

= 116.48 + 342.16 + 392 = $850.64

Federal unemployment = (0.006 × 2,080) + (0.006 × 6,110) + (0.006 × 7,000)

= 12.48 + 36.66 + 42 = $91.14

Option D. is the correct answer.

Answer:

80 degrees

Step-by-step explanation:

The entire circle represents 72 and the "slice of pie" is represented by a portion:

16/72 * 100

= 0.22 * 100

= 22.222%

.

22.222% of 360 degrees

= .22222 * 360

= 80 degrees

Each portion can be represented by 16/72

16/72 = 0.22

0.22... * 100% = 22.22...%

0.2222 * 360 = 80

Therefore, the answer is 80 degrees.

Best of Luck!

To find the x for a given function f(x) =5, use the graph of the function and search for places where the y-coordinate is 5. The x-coordinates of these points are the solutions.

The problem is asking for the value of x when f(x) = 5. To solve this problem, you would examine the graph of the function and look for the point(s) where the y-coordinate (the function value) is 5; the corresponding x-coordinate(s) would be your answer. For example, if seen directly above the number five on the y-axis, the line crosses at x=3, then x=3 is your solution. If it crosses again at x=-2, then x=-2 is another solution. Always remember that some problems may indeed have more than one answer, especially with functions that are not linear.

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Answer:

The inverse of h(x) is [tex]\frac{5x-6}{2}[/tex]

Step-by-step explanation:

* Lets explain how to make the inverse of a function

- To find the inverse of a function we switch x and y and then solve

  for new y

- You can make it with these steps

# write g(x) = y

# switch x and y

# solve for y

# write y as [tex]g^{-1}(x)[/tex]

* Lets solve the problem

∵ [tex]h(x)=\frac{2x+6}{5}[/tex]

# Step 1

∴ [tex]y=\frac{2x+6}{5}[/tex]

# Step 2

∴ [tex]x=\frac{2y+6}{5}[/tex]

# Step 3

∵ [tex]x=\frac{2y+6}{5}[/tex]

- Multiply each side by 5

∴ 5x = 2y + 6

- Subtract 6 from both sides

∴ 5x - 6 = 2y

- Divide both sides by 2

∴ [tex]y=\frac{5x-6}{2}[/tex]

# Step 4

∴ [tex]h^{-1}(x)=\frac{5x-6}{2}[/tex]

Answer: a) 0.0088

b) 0.2997

c)  5

Step-by-step explanation:

Given : Mean : [tex]\mu = 79[/tex] minutes

Standard deviation : [tex]\sigma = 8[/tex] minutes

The formula for z-score :

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

a) For x = 60 minutes

[tex]z=\dfrac{60-79}{8}=-2.375[/tex]

The p-value =[tex]P(z\leq-2.375)=0.0087745\approx0.0088[/tex]

b) For x = 75 minutes

[tex]z=\dfrac{75-79}{8}=-0.5[/tex]

The p-value =[tex]P(60<x<75)=P(-2.375<z<-0.5)[/tex]

[tex]=P(-0.5)-P(-2.375)=0.3085-0.0088=0.2997[/tex]

c) For x = 90 minutes

[tex]z=\dfrac{90-79}{8}=1.375[/tex]

The p-value =[tex]P(z>1.375)=1-P(z<1.375)[/tex]

[tex]=1-0.9154342=0.0845658[/tex]

If the number of students in the class = 60 .

Then , the number of students will be unable to complete the exam in the allotted time =[tex]0.0845658\times60=5.073948\approx5[/tex]

The probability of completing the exam in one hour or less is 0.0087. The probability that the exam is completed in more than 60 minutes but less than 75 minutes is 0.2998. We expect about 5 students to not finish the exam in the given 90 minutes.

In statistics, when a data set is normally distributed, we use a z-score to describe the position of a raw score in terms of its distance from the mean, when measured in standard deviation units. The formula to calculate a z-score is Z = (X - μ) / σ, where X is the raw score, μ is the mean, and σ is the standard deviation.

To answer the questions:

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Answer:

The selling price is $46 approx.

Step-by-step explanation:

The cost price is = $35.87

Markup is 22% of selling price.

Let the selling price be = 100%

So, 100%-22%=78% = 0.78

So, solution is = [tex]35.87/0.78=45.98[/tex] ≈ $46.

Hence, the selling price is $46 approx.

Answer:

We will accept the null hypothesis

Step-by-step explanation:

Given :[tex]H_0: \mu= 20[/tex]

          [tex]H_a: \mu> 20[/tex]

To Find :  What conclusion is appropriate in each of the following situations?

Solution :

z = 3.3, α = 0.05

So, first we will find the p value corresponding to z value in the z table

So, p-value is 0.999

Since p value is greater than Alpha

0.999>0.05

So, we will accept the null hypothesis

So, the population mean is 20

Answer:

"Decreasing the sample size."

Step-by-step explanation:

The margin of error is :

[tex]ME=z\times \frac{\sigma }{\sqrt{n}}[/tex]

Therefore, as n decreases, margin of error increases.

In the question we are asked " Which action will not reduce the margin of error."

Therefore, the correct option for which action will not reduce the margin of error is "Decreasing the sample size."

Answer: 0.2195

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 : The probability of that the general population has blood type A = [tex]\dfrac{1}{3}[/tex]

Sample size : n=6

Now, the probability that exactly three of them have blood type A is given by :-

[tex]P(3)=^6C_3(\dfrac{1}{3})^3(1-\dfrac{1}{3})^{6-3}\\\\=\dfrac{6!}{3!3!}(\dfrac{1}{3})^3(\dfrac{2}{3})^{3}\\\\=0.219478737997\approx0.2195[/tex]

Therefore, the probability that exactly three of them have blood type A = 0.2195

Answer:

(0.062, 0.130)

Step-by-step explanation:

Sample size = n = 500

Number of heads that were unemployed = x = 48

Proportion of heads that were unemployed = p = [tex]\frac{x}{n}=\frac{48}{500}=0.096[/tex]

Proportion of heads that were not unemployed = q = 1 - p = 1 - 0.096 = 0.904

Confidence Level = 99%

z-value for 99% confidence level = z = 2.58

The confidence interval about a population proportion is calculated as:

[tex](p-z\sqrt{\frac{pq}{n}} , p+z\sqrt{\frac{pq}{n}})[/tex]

Using the values, we get:

[tex](0.096-2.58\sqrt{\frac{0.096 \times 0.904}{500}},0.096+2.58\sqrt{\frac{0.096 \times 0.904}{500}})\\\\ = (0.062,0.130)[/tex]

Thus, 99% confidence interval for the proportion of unemployed heads of households in the population is (0.062, 0.130)

The 99% confidence interval for the proportion of unemployed heads of households in the population is approximately 0.096 ± 0.029.

To compute the 99% confidence interval for the proportion of unemployed heads of households, we can use the formula:

Confidence interval = sample proportion ± margin of error

1. Find the sample proportion:

Divide the number of unemployed heads of households (48) by the total number of heads of households surveyed (500).

Sample proportion = 48 / 500 = 0.096

2. Calculate the margin of error:

The margin of error depends on the level of confidence and the sample size. For a 99% confidence level, we need to find the critical value, which corresponds to 99% confidence and 500 as the sample size.

The critical value for a 99% confidence level and 500 as the sample size is approximately 2.576.

Margin of error = critical value * sqrt((sample proportion * (1 - sample proportion)) / sample size)

Margin of error = 2.576 * sqrt((0.096 * (1 - 0.096)) / 500) ≈ 0.029

3. Calculate the confidence interval:

Confidence interval = sample proportion ± margin of error

Confidence interval = 0.096 ± 0.029

For more such information on: confidence interval

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Answer:

Step-by-step explanation:

Based on the question we are given the percentages of each of the types of candies in the bag except for brown. Since the sum of all the percentages equals 75% and brown is the remaining percent then we can calculate that brown is (100-75 = 25%) 25% of the bag. Now we can show the probabilities of getting a certain type of candy by placing the percentages over the total percentage (100%).

Assuming the ratios/percentages of the candies stay the same having an infinite amount of candy will not affect the probabilities. That being said in order to calculate consecutive probability of getting 3 of a certain type in a row we have to multiply the probabilities together. This is calculated by multiplying the numerators with numerators and denominators with denominators.

[tex]\frac{15*85*15}{100*100*100} = \frac{19,125}{1,000,000} = \frac{1.9125}{100}[/tex]

[tex]\frac{80*80*80}{100*100*100} = \frac{512,000}{1,000,000} = \frac{51.2}{100}[/tex]

[tex]\frac{20*100*100}{100*100*100} = \frac{200,000}{1,000,000} = \frac{20}{100}[/tex]

Given:

Probability of winning, P(X) = 0.02

Probability of losing, P([tex]\bar{X}[/tex]) = 0.98

Wining amount = $160

Losing amount = $16

Step-by-step explanation:

Let the expected amount of money win be  'X'

Expected value of X, E(X) = Probability of winning, P(X).Probability of winning, P(X)  - Probability of losing, P([tex]\bar{X}[/tex]).Losing amount

Now,

E(X) = ([tex]0.02\times 160 - 0.98\times 16[/tex])

E(X) = -12.48

Expected value of X = -12.48

Expected loss value = $12.48 loss

Answer:

[tex]x=1+\dfrac{3}{1-Ke^{3t}}[/tex]

Step-by-step explanation:

Given that

[tex]\dfrac{dx}{dt}=x^2-5x+4[/tex]

This is a differential equation.

Now by separating variables

[tex]\dfrac{dx}{x^2-5x+4}=dt[/tex]

[tex]\dfrac{dx}{(x-1)(x-4)}=dt[/tex]

[tex]\dfrac{1}{3}\left(\dfrac{1}{x-4}-\dfrac{1}{x-1}\right)dx=dt[/tex]

Now by integrating both side

[tex]\int\dfrac{1}{3}\left(\dfrac{1}{x-4}-\dfrac{1}{x-1}\right)dx=\int dt[/tex]

[tex]\dfrac{1}{3}\left(\ln(x-4)-\ln(x-1)\right )=t+C[/tex]

Where C is the constant

[tex]\dfrac{1}{3}\ln\dfrac{x-4}{x-1}=t+C[/tex]

[tex]\dfrac{x-4}{x-1}=Ke^{3t}[/tex]    K is the constant.

[tex]x=1+\dfrac{3}{1-Ke^{3t}}[/tex]

So the solution of above differential equation is

[tex]x=1+\dfrac{3}{1-Ke^{3t}}[/tex]

Answer:

A. 15 and 0. B. 15 and 17

Step-by-step explanation:

A. a= 13 and b = 195.

b div a = 195 div 13. The result is the integer part of the division, so

195 div 13 = 15.

b mod a = 195 mod 13. The result is the residue of the division. In this case the division is exact, so

195 mod 13 = 0.

B. a = 24 and b=377.

b div a = 377 div 24. The result is the integer part of the division, so

377 div 24 = 15.

b mod a = 377 mod 24. The result is the residue of the division. In this case the division is not exact, so

377 = 24*15+17, then

377 mod 24 = 17.

Answer:

g(-0.5) = -1

g(0.2) = 0

g(0.5) = 1

Step-by-step explanation:

We are given the value of g(x) for which the x is defined.

Solving

g(-0.5) = -1

As given g(x) = -1 if -1.5 ≤ x ≤ 0.5

g(0.2) = 0

As given g(x) = 0 if -0.5 < x < 0.5

g(0.5) = 1

As given g(x) = 1 if 0.5 ≤ x < 1.5

Answer: 465

Step-by-step explanation:

251 (days) x 24 (hours) = 6,024 hours

6,024+21 hours= 6,045

6045/13=465

Answer:

  m∠1=1/2(mAB+mEF)

Step-by-step explanation:

The measure of the angle is half the sum of the intercepted arcs.

Answer:

[tex]m\angle 1 = \frac{1}{2}(m\angle AB+m\angle EF)[/tex]

Step-by-step explanation:

When two chords intersect each other inside a circle, the measure of the angle formed is one half the sum of the measure of the intercepted arcs.

Here, the chords FA and BE intersected each other inside the circle,

Also, angle 1 is the angle formed by the intersection,

Thus, from the above statement,

[tex]m\angle 1 = \frac{1}{2}(m\angle AB+m\angle EF)[/tex]

Second option is correct.

Answer:

The center require to have a 4.6 inspectors, but this on you decide if you have 4 or 5 inspectors, it may not have 4.6 people working.

So then if you decide to have 5 people their utilization percentage is 83.3%, but if you decide to have 4 people their utilization rate will be 104.17% at the risk of defaulting on demand.

Step-by-step explanation:

1. Define your variables

Demand rate= 50 vehicles per hour

Service rate = 12 vehicles per hour per inspector

Inspectors= a

2. Use the formule of the center´s utilization

U= [tex]\frac{Demand}{(Service) X a}[/tex]

0,9= [tex]\frac{50}{(12) X a }[/tex]

0,9a=[tex]\frac{50}{12}[/tex]

a=[tex]\frac{50}{12X0,9}[/tex]

a= 4.6

3. According to the center´s utilization formula the center require 4,6 inspectors, but approaches 5 because people cannot be divided. With this numbers of inspectors the utilization is:

U= [tex]\frac{Demand}{(Service) X (No. inspectors)}[/tex]

U= [tex]\frac{50}{(12) X (5)}[/tex]

U= 83,3%

4. Other option that you will be used is that the center require 4 inspectors. With this numbers of inspectors the utilization is:

U= [tex]\frac{Demand}{(Service) X (No. inspectors)}[/tex]

U= [tex]\frac{50}{(12) X (4)}[/tex]

U= 104,17%

The center would require 6 inspectors to have a target utilization of 90 percent.

To determine the number of inspectors required to have a target utilization of 90 percent, we can use the formula:

Number of inspectors = Total vehicles per hour / Vehicles inspected per inspector per hour / Target utilization

Given that the center tests 50 vehicles per hour, and each inspector can inspect 12 vehicles per hour, the formula becomes:

Number of inspectors = 50 / 12 / 0.9 = 5.56

Since we cannot have a fraction of an inspector, we round up to the nearest whole number. Therefore, the center would require 6 inspectors to have a target utilization of 90 percent.

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Answer:

[tex]y=3e^{-4t}[/tex]

Step-by-step explanation:

[tex]y''+5y'+4y=0[/tex]

Applying the Laplace transform:

[tex]\mathcal{L}[y'']+5\mathcal{L}[y']+4\mathcal{L}[y']=0[/tex]

With the formulas:

[tex]\mathcal{L}[y'']=s^2\mathcal{L}[y]-y(0)s-y'(0)[/tex]

[tex]\mathcal{L}[y']=s\mathcal{L}[y]-y(0)[/tex]

[tex]\mathcal{L}[x]=L[/tex]

[tex]s^2L-3s+5sL-3+4L=0[/tex]

Solving for [tex]L[/tex]

[tex]L(s^2+5s+4)=3s+3[/tex]

[tex]L=\frac{3s+3}{s^2+5s+4}[/tex]

[tex]L=\frac{3(s+1)}{(s+1)(s+4)}[/tex]

[tex]L=\frac3{s+4}[/tex]

Apply the inverse Laplace transform with this formula:

[tex]\mathcal{L}^{-1}[\frac1{s-a}]=e^{at}[/tex]

[tex]y=3\mathcal{L}^{-1}[\frac1{s+4}]=3e^{-4t}[/tex]

Answer: 0.5467

Step-by-step explanation:

Let X be the random variable that represents the income (in dollars) of a randomly selected person.

Given : [tex]\mu=15451[/tex]

[tex]\sigma^2=298116\\\\\Rightarrow\ \sigma=\sqrt{298116}=546[/tex]

Sample size : n=350

z-score : [tex]\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

To find the probability that the sample mean would differ from the true mean by less than 22 dollars, the interval will be

[tex]\mu-22,\ \mu+22\\\\=15,451 -22,\ 15,451 +22\\\\=15429,\ 15473[/tex]

For x=15429

[tex]z=\dfrac{15429-15451}{\dfrac{546}{\sqrt{350}}}\approx-0.75[/tex]

For x=15473

[tex]z=\dfrac{15473-15451}{\dfrac{546}{\sqrt{350}}}\approx0.75[/tex]

The required probability :-

[tex]P(15429<X<15473)=P(-0.75<z<0.75)\\\\=1-2(P(z<0.75))=1-2(0.2266274)=0.5467452\approx0.5467[/tex]

Hence, the required probability is 0.5467.

Assume a solution of the form [tex]\Psi(x,y)=C[/tex]. Differentiating both sides gives

[tex]\Psi_x\,\mathrm dx+\Psi_y\,\mathrm dy=0[/tex]

with [tex]\Psi_x=y[/tex] and [tex]\Psi_y=x(\ln x-\ln y-1)[/tex].

Divide both sides by [tex]x[/tex] and we have

[tex]\dfrac yx\,\mathrm dx+(\ln x-\ln y-1)\,\mathrm dy=0[/tex]

Notice that

[tex]\left(\dfrac yx\right)_y=\dfrac1x[/tex]

[tex]\left(\ln x-\ln y-1\right)_x=\dfrac1x[/tex]

so the ODE is exact. Now we can look for a solution [tex]\Psi[/tex] with

[tex]\Psi_x=\dfrac yx[/tex]

[tex]\Psi_y=\ln x-\ln y-1[/tex]

Integrating the first PDE with respect to [tex]x[/tex] gives

[tex]\Psi(x,y)=y\ln x+f(y)[/tex]

and differentiating this with respect to [tex]y[/tex] gives

[tex]\Psi_y=\ln x+f'(y)=\ln x-\ln y-1\implies f'(y)=-\ln y-1\implies f(y)=-y\ln y+C[/tex]

So this ODE has general solution

[tex]y\ln x-y\ln y=C[/tex]

Given that [tex]y(1)=e[/tex], we have

[tex]e\ln1-e\ln e=C\implies C=-e[/tex]

so the particular solution is

[tex]y(\ln x-\ln y)=-e[/tex]

[tex]y\ln\dfrac xy=-e[/tex]

[tex]\boxed{y\ln\dfrac yx=e}[/tex]