The amounts of time that customers stay in a certain restaurant for lunch is normally distributed with a standard deviation of 17 minutes. A random sample of 50 lunch customers was taken at this restaurant. Construct a 99% confidence interval for the true average amount of time customers spend in the restaurant for lunch. Round your answers to two decimal places and use ascending order.

Compute the volume of [tex]Q[/tex]:

[tex]\displaystyle\iiint_Q\mathrm dV=\int_0^2\int_0^{2-x}\int_0^{2-x-y}\mathrm dz\,\mathrm dy\,\mathrm dx=\frac43[/tex]

Integrate [tex]f(x,y,z)=x+y+z[/tex] over [tex]Q[/tex]:

[tex]\displaystyle\iiint_Qf(x,y,z)\,\mathrm dV=\int_0^2\int_0^{2-x}\int_0^{2-x-y}(x+y+z)\,\mathrm dz\,\mathrm dy\,\mathrm dx=2[/tex]

So the average value of [tex]f[/tex] over [tex]Q[/tex] is 2/(4/3) = 3/2.

To solve this mathematical problem, we need to understand the Average Value of a Continuous function.

The average value of a continuous function is derived by taking the integral of the function over the interval. This is then divided using the length of that interval.

To determine the average value of the function  f(x, y, z), over the  solid region named Q,

we can say:

[tex]\int\int\int _{Q}[/tex]  dV = [tex]\int_{0}^{2} \int_{0}^{2-x} \int_{0}^{2-x-y}[/tex]  dzdydx = 4/3

Integrating the above, we have

[tex]\int\int\int _{Q}[/tex] [tex]f(x,y,z)[/tex] dV = [tex]\int_{0}^{2} \int_{0}^{2-x} \int_{0}^{2-x-y}[/tex]   (x+ y + z) dzdydx  = 2

Therefore, the average value of the function f over the Solid region Q becomes:

2/ (4/3) = 1.5 or 3/2

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Answer: No, the Volume is x^3 - 3x^2 - 10x

Step-by-step explanation:

Since the volume of the cubic storage unit is x^3

Therefore,

Length = x

Width = x

Height = x

For the new storage unit

Length = x + 2

Width = x

Height = x - 5

Volume = ( x + 2)(x)(x -5)

V = x (x^2 -3x - 10)

V = x^3 - 3x^2 - 10x

Therefore, the volume of the new storage unit is x^3 - 3x^2 - 10x

Answer:the customer is incorrect

Step-by-step explanation:

In a cube, all 4 sides are equal. The volume of a cube that has x as the length of each side would be x^3

If the customer thinks that f the volume of the cube is x^3, it means that each side is x. Then the other storage unit offered to the customer is 2 feet longer,5 feet shorter than the first unit. Its dimensions would be (x+ 2) feet, (x - 5) feet and x feet

The volume of the other storage unit should be

x[(x + 2)(x - 5)] = x(x^2 - 5x + 2x + 10)

= x(x^2 - 3x + 10)

= x^3 - 3x^2 + 10x

Answer:

Option B) In control as this one data point is not more than three standard deviations from the mean

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 104 rotations per minute

Standard Deviation, σ = 8.2 rotations per

We are given that the distribution of process is a bell shaped distribution that is a normal distribution.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

For x = 118

[tex]z = \displaystyle\frac{118-104}{8.2} = 1.7073[/tex]

Thus, we could say that this data point lies within three standard deviations from the mean as:

[tex]\mu - 3\sigma < x < \mu + 3\sigma\\104-3(8.2) < x < 104 + 3(8.2)\\79.4 < 118 < 128.6[/tex]

Thus, it could be said

Option B) In control as this one data point is not more than three standard deviations from the mean

The process would be considered out of control as the randomly selected minute has more than two standard deviations away from the mean.

To determine whether the process is in control or out of control, we can use the Empirical Rule. The Empirical Rule states that approximately 68 percent of the data is within one standard deviation of the mean, approximately 95 percent of the data is within two standard deviations of the mean, and more than 99 percent of the data is within three standard deviations of the mean. In this case, since the randomly selected minute has 118 rotations per minute, which is more than two standard deviations away from the mean (104 rotations per minute), the process would be considered out of control.

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Answer:

[tex]ME= \frac{Width}{2}=\frac{0.078}{2}=0.039[/tex]

[tex]\hat p =0.688+0.039=0.727[/tex]

[tex]\hat p =0.766-0.039=0.727[/tex]

Step-by-step explanation:

Assuming that the confidence interval is (0.688; 0.766)

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

The population proportion have the following distribution  

[tex]p \sim N(p,\sqrt{\frac{p(1-p)}{n}})[/tex]  

The confidence interval would be given by this formula  

[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]  

For the 95% confidence interval the value of [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2=0.025[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.  

[tex]z_{\alpha/2}=1.96[/tex]  

Use the confidence interval to find the point estimate and margin of error for the proportion

The margin of error is given by :

[tex]Me=z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

And for our case we can find the width of the confidence interval like this:

Width =0.766-0.688=0.078

And the estimation for the margin of error would be given by:

[tex]ME= \frac{Width}{2}=\frac{0.078}{2}=0.039[/tex]

Now we can find th point of estimate adding the margin of error to the lower limit of the interval or subtracting the margin of error to the upper limit, like this:

[tex]\hat p =0.688+0.039=0.727[/tex]

[tex]\hat p =0.766-0.039=0.727[/tex]

The point estimate is the midpoint of the confidence interval, calculated using the equation: (a + b) / 2. The margin of error is the amount by which the point estimate could differ from the actual proportion, calculated as the absolute difference between the point estimate and either limit of the confidence interval.

The question doesn't provide a specific confidence interval. Hence, let's take it in a general form as (a, b). Here, a and b are the lower and upper limits of the 95% confidence interval for the proportion of customers interested in a breakfast menu.

As per the properties of confidence intervals, the point estimate is the midpoint of the interval. It is calculated as the sum of the lower and upper limits divided by 2.

To calculate this: Point estimate = (a + b) / 2

On the other hand, the margin of error is the distance from this point estimate to either of the confidence interval limits (upper or lower). It can be calculated as the difference between the point estimate and the lower limit (or the upper limit).

To calculate this: Margin of Error = |Point estimate - a| = |Point estimate - b|

If you have specifics for 'a' and 'b', you can substitute those values into the point estimate and margin of error equations to get those values precise to three decimal places.

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Answer: B. 0.036

Step-by-step explanation:

Formula for standard error :

[tex]SE=\sqrt{\dfrac{p(1-p)}{n}}[/tex]

, where p = Population proportion and n= sample size.

Let p be the population proportion of the people who favor new taxes.

As per given , we have

n= 170

[tex]p=\dfrac{55}{170}\approx0.324[/tex]

Substitute these values in the formula, we get

[tex]SE=\sqrt{\dfrac{0.324(1-0.324)}{170}}\\\\=\sqrt{0.00129}\\\\=0.0359165699921\approx0.036[/tex]

Hence, the standard error of the estimate is 0.036.

∴ The correct answer is OPTION B. 0.036

Answer:

b. severs the corpus callosum between hemispheres.

Step-by-step explanation:

The split-brain surgery is used to alleviate epileptic seizures. It involves the severing of the corpus callosum, that is the bond between both hemispheres of the brain.

So the correct answer is:

b. severs the corpus callosum between hemispheres.

Answer:

$173493.8

Step-by-step explanation:

Data provided in the question:

LSRL for the data is ? = 3.8785x + 18.3538

Here,

x is area in 100 square feet

and

price in thousands of dollar

Thus,

For the given area 4000 square foot

x = 4000 ÷ 100 = 40                  [Area in 100 square feet]

Therefore,

Using the given equation

Price  = 3.8785(40) + 18.3538

or

Price = 173.4938 in thousands of dollar

or

Price = 173.4938 × $1000

Price = $173493.8

The test and evaluation (T&E) product that is required will be

It should be noted that a Test and Evaluation Master Plan is used as the planning and management tool for the test activities.

On the other hand, Milestone B is assumed as the official start of a program. It's a MDA led review at the final phase of the Technology Maturation and Risk Reduction.

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Among the options provided, the correct answer is B. Operational Assessment, as it is a required T&E product at Milestone B.

In the Department of Defense Acquisition Process, Milestone B represents a significant point in the acquisition lifecycle where programs are reviewed and approved for entry into the Engineering and Manufacturing Development (EMD) phase.

At Milestone B, the primary focus is on ensuring that the program is mature enough to proceed into EMD with an acceptable level of risk and that adequate planning has taken place.

Among the options provided:

A. Waiver of Military Equipment Program Description: This document is typically not required at Milestone B. It may be relevant in other phases but not a direct requirement at this milestone.

B. Operational Assessment: Operational Assessment (OA) is a crucial T&E product at Milestone B. It assesses the operational effectiveness and suitability of the system under realistic operational conditions, helping to determine whether the system is suitable for further development and production.

C. Identification of LRIP Quantities: The identification of Low-Rate Initial Production (LRIP) quantities is also an important consideration at Milestone B. Decisions about LRIP quantities are critical for moving forward with limited production to further test and refine the system.

D. No T&E products are required at MS B: This statement is not accurate. Milestone B does require specific T&E products, such as the Operational Assessment and considerations related to LRIP quantities, as mentioned above.

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Answer:

n=1041  or higher

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

The population proportion have the following distribution

[tex]p \sim N(p,\sqrt{\frac{\hat p(1-\hat p)}{n}})[/tex]

2) Solution to the problem

In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 99% of confidence, our significance level would be given by [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2 =0.005[/tex]. And the critical value would be given by:

[tex]z_{\alpha/2}=-2.58, z_{1-\alpha/2}=2.58[/tex]

The margin of error for the proportion interval is given by this formula:  

[tex] ME=z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]    (a)  

And on this case we have that [tex]ME =\pm 0.04[/tex] and we are interested in order to find the value of n, if we solve n from equation (a) we got:  

[tex]n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}[/tex]   (b)  

Since we don't have a prior estimation for th proportion of interest, we can use this value as an estimation [tex]\hat p =0.5[/tex] And replacing into equation (b) the values from part a we got:

[tex]n=\frac{0.5(1-0.5)}{(\frac{0.04}{2.58})^2}=1040.06[/tex]  

And rounded up we have that n=1041  or higher.

Answer:

Step-by-step explanation:

Answer:

a) during the first 10 hours of growth the weight will increase 271.8% relative to the initial state ( or 2.718 g)

b) during the first 20 hours of growth the weight will increase  467.1 % relative to the initial state ( or 4.671 g)

Step-by-step explanation:

since the growing rate law is

W'(t) = 0.1 gr/hour * e^(0.1gr/hour* t)  , W'(t) [gr/hour]

and following mathematical conventions: W'(t)= dW/dt

then

dW/dt=0.1 e^(0.1t)

∫dW =∫0.1 e^(0.1t) dt

W = e^(0.1t) + C

at the beginning, (time t=0) the weight is W=2 grams .Therefore

2 g = e^(0.1 g/h*0) + C  → 2 g = 1 g + C → C = 1 g

then

W = e^(0.1t) + 1 g

at t= 10 hours

W = e^(0.1 g/h*10h) + 1 g = 3.718 g/h

therefore the weight will increase

ΔW = 3.718 g -  1 g = 2.718 g  or 271.8% relative to the initial state

for t=20 hours

W = e^(0.1 g/h*20h) + 1 g = 8.389 g/h

thus, the from t= 10 hours to t= 20 hours the weight will increase

ΔW =  8.389 g/h -  3.718 g = 4.671 g  or 467.1 %relative to the initial state

Final answer:

To find the growth of the yeast culture, we integrate the growth rate function over the respective intervals. For the first 10 hours, the yeast weight increases by roughly 1.718 grams. From the 10th to the 20th hour, it increases by approximately 4.671 grams.

Explanation:

To do this, we integrate the provided growth rate function over the given time intervals. Initially, the yeast culture weighs 2 grams and grows at the rate of W'(t) = 0.1e0.1t grams per hour.

Finding the Weight Increase During the First 10 Hours

To find the total growth over the first 10 hours, we calculate the integral of the growth rate from t = 0 to t = 10.

[tex]\[ \int_{0}^{10} 0.1e^{0.1t} dt = \left[ \frac{0.1}{0.1}e^{0.1t} \right]_{0}^{10} = \left[ e^{0.1t} \right]_{0}^{10} = e^{1} - e^{0} = e - 1 \][/tex]

Since e is approximately 2.718, this value becomes approximately 2.718 - 1, which equals 1.718 grams. Hence, during the first 10 hours, the yeast culture will increase by about 1.718 grams.

Weight Increase from the End of the 10th to the End of the 20th Hour

Again, we integrate the growth rate, but this time from t = 10 to t = 20.

[tex]\[ \int_{10}^{20} 0.1e^{0.1t} dt = \left[ e^{0.1t} \right]_{10}^{20} = e^{2} - e^{1} \][/tex]

Similar to the previous calculation, this yields e² - e which is approximately 7.389 - 2.718, resulting in an increase of approximately 4.671 grams.

Answer:

4×10⁶

Step-by-step explanation:

Concept to know is that when you divide two numbers with the same base, you subtract their exponent.

So this problem could be split into 2 parts. The non power-of-ten numbers, and the power of tens numbers.

I divided 9.2/2.3 and got 4.

I then divided 10⁸/10² and got 10⁸⁻² = 10⁶.

Put it together and you get 4×10⁶

Number of times the value of 2.3×10² from 9.2×10⁸ is 4×10⁶.

Scientific notation is a method for expressing a given quantity as a number having significant digits necessary for a specified degree of accuracy, multiplied by 10 to the appropriate power such as 1.56×10⁷.

The given numbers are 9.2×10⁸ and 2.3×10².

Now, number of times =9.2×10⁸/2.3×10²

= 4×[tex]10^{8-2}[/tex]

= 4×10⁶

Therefore, number of times the value of 2.3×10² from 9.2×10⁸ is 4×10⁶.

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Answer:

[tex]t=\frac{23500-20000}{\frac{3900}{\sqrt{100}}}=8.974[/tex]      

[tex]p_v =P(t_{99}>8.974)=9.43x10^{-15}[/tex]  

If we compare the p value and the significance level given for example [tex]\alpha=0.05[/tex] we see that [tex]p_v<<\alpha[/tex] so we can conclude that we to reject the null hypothesis, and the actual mean is significantly higher than 20000.      

Step-by-step explanation:

1) Data given and notation      

[tex]\bar X=23500[/tex] represent the sample mean  

[tex]s=3900[/tex] represent the sample standard deviation      

[tex]n=100[/tex] sample size      

[tex]\mu_o =2000[/tex] represent the value that we want to test    

[tex]\alpha[/tex] represent the significance level for the hypothesis test.    

t would represent the statistic (variable of interest)      

[tex]p_v[/tex] represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.      

We need to conduct a hypothesis in order to determine if the true mean is higher than 20000, the system of hypothesis would be:      

Null hypothesis:[tex]\mu \leq 2000[/tex]      

Alternative hypothesis:[tex]\mu > 2000[/tex]      

We don't know the population deviation, so for this case is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:      

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)      

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic      

We can replace in formula (1) the info given like this:      

[tex]t=\frac{23500-20000}{\frac{3900}{\sqrt{100}}}=8.974[/tex]      

Calculate the P-value      

First we need to calculate the degrees of freedom given by:  

[tex]df=n-1=100-1=99[/tex]  

Since is a one-side upper test the p value would be:      

[tex]p_v =P(t_{99}>8.974)=9.43x10^{-15}[/tex]  

Conclusion      

If we compare the p value and the significance level given for example [tex]\alpha=0.05[/tex] we see that [tex]p_v<<\alpha[/tex] so we can conclude that we to reject the null hypothesis, and the actual mean is significantly higher than 20000.      

To evaluate the claim that automobiles are driven on average more than 20,000 kilometers per year, a one-sample t-test has to be conducted using the sample data. Based on the outcome of the test and the P-value generated, we can either reject or fail to reject the null hypothesis, hence determining whether to agree with the claim.

To determine whether you'd agree with the claim that automobiles are driven on average more than 20,000 kilometers per year based on the sample data, you would need to perform a hypothesis test. In this case, the null hypothesis (H0) would be that the average distance driven is equal to 20,000 kilometers. The alternative hypothesis (H1) would be that the average distance driven is more than 20,000 kilometers.

Your sample data indicates an average of 23,500 kilometers with a standard deviation of 3900 kilometers. To conduct the hypothesis test, you can use the formulas for a one-sample t-test because the population standard deviation is unknown. The P-value generated from the test will tell you whether to reject the null hypothesis.

If the P-value is less than the chosen significance level (commonly 0.05), you would reject the null hypothesis and conclude that automobiles are driven, on average, more than 20,000 kilometers per year. Otherwise, you would fail to reject the null hypothesis and could not conclusively agree with the claim. Without knowing the exact P-value derived from the test, it would be impossible to definitively agree or disagree with the claim based on the provided sample data.

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The particle's position, represented by s(t), is found from its velocity v(t) using integration. The function of the velocity is rewritten, integrated, and a constant of integration is found using the given initial condition.

The problem given involves a particle moving with a certain velocity function, v(t) = 1.5√t, and an initial position at t = 4, that is, s(4) = 14. The problem asks for the position of the particle, which is often represented by a displacement or position function, denoted commonly as s(t).

To solve this problem, we need to use the fundamental relationship between velocity and position, which states that velocity is the rate of change of position with respect to time. This relationship implies that to find the position function from the velocity function, we need to find an antiderivative or integral of the velocity function.

By following this step-by-step method, we can determine the position of the particle based on the given information.

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The position of the particle is given by the function s(t) = t^(3/2) + 6.

To find the position of the particle, we need to integrate the given velocity function, [tex]v(t)=1.5\sqrt{t}[/tex].

First, let's find the indefinite integral of v(t):

[tex]\int\limits {v(t)} \, dt = \int\limits {1.5\sqrt{t} } \, dt[/tex]

Rewrite the square root as a power:

[tex]=\int\limits {1.5t^{\frac{1}{2} } \, dt[/tex]

Apply the power rule of integration:

[tex]=\frac{1.5t^{(\frac{1}{2}+1 )} }{\frac{1}{2} +1} + C[/tex]

[tex]= 1.5*(\frac{2}{3} )t^{\frac{3}{2}}+C[/tex]

[tex]=t^\frac{3}{2} +C[/tex]

Now we use the initial condition s(4) = 14 to solve for C:

Therefore:

So the position function is:

[tex]s(t)=t^\frac{3}{2} +6[/tex]

Work is provided in the image attached.

Answer:

d) 35

Step-by-step explanation:

Consider the venn diagram attached below

Given

n(U) = 200

n(A) = 165

n(B) = 95

n(A ∩ B ) = 80

n([tex]A^{c}[/tex] ∪ B) =?

Using

n(A  ∪ B) = n(A) + n(b) - n(A ∩)B

For [tex]A^{c}[/tex]

[tex]n(A^{c}\cup B) = n(A^{C})+ n(B)-n(A^{c}\cap B)---(1)\\n(A^{c})=U-A\\n(A^{c})=n(U)-n(A)\\A^{c}\cap B=B\\n(A^{c}\cap B) =n(B)\\[/tex]

Then (1) becomes

[tex]n(A^{c}\cup B) = n(U)-n(A)+ n(B)-n(B)\\n(A^{c}\cup B)=200-165+95-95\\n(A^{c}\cup B)=35[/tex]

Your Answer Should Be

3m+6=24

Answer:

Step-by-step explanation:

Let x represent the number of trips to the airport and

Let y represent the number of trips from the airport.

A taxi driver had 44 fares to and from the airport last Monday. This means that

x + y = 44

The price for a ride to the airport is $7, and the price for a ride from the airport is $6. The driver collected a total of $289 for the day. This means that

7x + 6y = 289 - - - - - - - - - - - 1

Substituting x = 44 - y into equation 1, it becomes

7(44 - y) + 6y = 289

308 - 7y + 6y = 289

- 7y + 6y = 289 - 308

-y = - 19

y = 19

x = 44 - y

x = 44 - 19

x = 25

Answer:

(A) oo, ou, uo, uu

(B) 1/4

(C) 1/2

Step-by-step explanation:

(A) When using o for overweight and u for underweight, there are four possible weight outcomes which are; oo, ou, uo, uu

                   The sample space would be: oo, ou, uo, uu

This implies there are 4 possible outcomes.

(B) From the sample space, the event, getting two underweight weight children occurs only once, uu. The probability of getting two underweight children = 1/4

(C)  From the sample space, the event, getting one overweight child and one underweight child occurs twice, ou, uo.

    The probability of getting one overweight child and one underweight child = 2/4 = 1/2

Answer:

F = 7476 N

Step-by-step explanation:

given,

diameter of hemispherical plate = 6 ft

height of submergence = 1 ft

the weight density of water =  62.5 lb/ft³

Assuming that hemispherical plate is residing on x and y axis.

bottom of plate is on x-axis and left side of the plate touches y-axis

now, plate is defined by the upper half of the circle

(x - 3)² + (y-0)² = 3²

  y² = 9 - (x - 3)²

  y = √(9 - (x - 3)²)

hydro static pressure on one  side of plate.

[tex]F = \int \rho g x w(x)dx[/tex]

[tex]F = \int_0^3 62.5\times 9.8 x \times \sqrt{9-(x-3)^2}dx[/tex]

[tex]F = 612.5 \int_0^3 x \times \sqrt{9-(x-3)^2}dx[/tex]

on solving the above equation

[tex]F = 612.5(27\dfrac{\pi}{4}-9)[/tex]

F = 7476 N

Answer:

Option D -[tex]\frac{7}{8}\times 8\frac{1}{10}\approx 1\times 8[/tex]

Step-by-step explanation:

To find : Which expression is the best estimate of the product of [tex]\frac{7}{8}[/tex] and [tex]8\frac{1}{10}[/tex]?

Solution :

We estimate the number individually,

[tex]\frac{7}{8}=0.875[/tex]

Estimate the number we get 0.875≈1.

[tex]8\frac{1}{10}=\frac{81}{10}[/tex]

[tex]8\frac{1}{10}=8.1[/tex]

Estimate the number we get 8.1≈8.

The product of [tex]\frac{7}{8}[/tex] and [tex]8\frac{1}{10}[/tex] is

[tex]\frac{7}{8}\times 8\frac{1}{10}\approx 1\times 8[/tex]

Therefore, option D is correct.

Answer:

Flux across S = 72π

Step-by-step explanation:

First we need to calculate the divergence of the vector field:

Div F = [tex]\frac{dFx}{dx} + \frac{dfy}{dy} + \frac{dFz}{dz}[/tex]

Where

Fx = 8y^2 - 3x

Fy = -9x+4y

Fz = -2y^3 +z

Then

Div F = -3 +4 + 1  = 2

And how the vector field’s divergence is a constant, we can calculate the flux across of the surface how:

Flux across S = Div F * Volume of Sphere

Fluz acroos S = 2(4/3)π[tex]r^{3}[/tex]

                 r : Sphere’s radio

Flux across S = (2)(4/3)π[tex]3^{3}[/tex]

                       = 72π

To compute the net outward flux of the field across the sphere, first calculate the divergence of the field and then apply the Divergence Theorem. The flux equals the integral of the divergence over the volume of the sphere.

The first step of this problem is to compute the divergence of the vector field F. The divergence is the scalar quantity obtained by performing a dot product of the del operator with the field. For the given field F = <8y² - 3x, -9x + 4y, -2y³+ z>, the divergence is thus Div(F) = ∇.F = d(8y² - 3x)/dx + d(-9x + 4y)/dy + d(-2y³ + z)/dz.

For the sphere S where x² + y² + z² = 9, the radius r is √9 = 3. According to the Divergence Theorem, the flux across the boundary of the volume enclosed by S equals the triple integral of the divergence over the volume. So, the outward flux = ∫∫∫(Div(F).dV), where the triple integral is taken over the volume of the sphere.

Carry out the calculations to find the exact value of the outward flux.

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Answer: the principal is approximately 23377

Step-by-step explanation:

Let the Initial amount deposited into the account be $x This means that the principal is P = $x

It was compounded quarterly. This means that it was compounded four times in a year. So

n = 4

The rate at which the principal was compounded is 3.575%. So

r = 3.575/100 = 0.03575

It would be compounded for 30 years So

t = 30

The formula for compound interest is

A = P(1+r/n)^nt

A = total amount in the account at the end of t years.

A is given as $68,000

Therefore

68000= x (1+0.03575/4)^4×30

68000= x (1+0.0089375)^120

68000= x (1.0089375)^120

68000 = 2.90878547719x

x = 68000/2.90878547719

x = 23377.4545

Answer:

The answer is B.

Step-by-step explanation:

23,377.

Answer:

1,non linear and separable

Step-by-step explanation:

given is a differential equation as

[tex]\frac{dy}{dx} =y(y-2)e^x[/tex]

Here we have derivative highest is first derivative

I) Order = 1 (since first derivative is used)

2) It is not linear since the variable y has power 2.

3) To check whether separable or not

[tex]\frac{dy}{dx} =y(y-2)e^x[/tex]

we can take all y variables to left side and x to right side

Hence separable

Answer:

-a discrete random variable

Step-by-step explanation:

The number of customers that enter a store can be 0,1,2,...,100,1000,etc...

If cannot be a decimal number, for example, 0.5. So it is a discrete random variable.

The correct answer is:

-a discrete random variable

The solution is a three-step process. First, solve the characteristic equation and determine the homogeneous solution. Second, use the method of undetermined coefficients to find the particular solution. Finally, the general solution to the nonhomogeneous differential equation equals the sum of the homogeneous solution and the particular solution.

To solve this, we'll need to go through three stages: solving the homogeneous equation, finding the particular solution, and finally combining these to form the general solution.

Step 1: The associated homogeneous equation is y'' + 4y' + 5y = 0. The general solution to this homogeneous equation can be obtained by solving the characteristic quadratic equation r^2 + 4r + 5 = 0. You will find that the roots are complex, and the general solution for the homogeneous differential equation would be in the form yh = c1*e^(-2x)cos(x)+c2*e^(-2x)sin(x).

Step 2: The particular solution of the nonhomogeneous differential equation can be obtained using the method of undetermined coefficients or the method of variation of parameters. For this case, we will use the method of undetermined coefficients. You will eventually find after performing these methods that the particular solution yp is of the form yp = Ax + Be^(?x), where A and B are constants which can be calculated.

Step 3: Finally, the general solution to the nonhomogeneous differential equation we are trying to solve is simply the sum of the general solution from Step 1 (the homogeneous solution) and the particular solution from Step 2. This would yield a solution y = c1*e^(-2x)cos(x) + c2*e^(-2x)sin(x) + Ax + Be^(?x).

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The solution to the homogeneous differential equation is yh=c1e^-2x cos(x) + c2e^-2x sin(x). The particular solution, yp, can be obtained using the method of undetermined coefficients for the nonhomogeneous part, -10x+e^(-x). The final solution y is the sum of yh and yp.

The nonhomogeneous differential equation you have provided is in the form of y″+4y′+5y=-10x+e^(-x).

To find the particular solution (yp), we first need to find the homogeneous solution. The characteristic equation of the associated homogeneous differential equation is r^2+4r+5=0. Solving this quadratic equation, we obtain complex roots as r = -2±i. Hence, the homogeneous solution (yh) is expressed as yh= c1e^-2x cos(x) + c2e^-2x sin(x)

Next, to find yp for f(x)= -10x+e^(-x), we use the method of undetermined coefficients. However, due to the limitation of the platform, it would be too complicated to carry out this procedure here.

Eventually, the final solution of the original nonhomogeneous differential equation will be obtained by adding yh and yp, i.e., y= yh+yp.

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Answer:

The smallest integer is 9 and there are 19 terms in the sequence.

Step-by-step explanation:

Arithmetic Sequence

The general term of an arithmetic sequence is

[tex]\displaystyle a_n=a_1+(n-1)r\ ........[eq\ 1][/tex]

And the sum of all n terms is

[tex]\displaystyle s_n=\frac{a_1+a_n}{2}n...... [eq\ 2][/tex]

The sequence of the question complies with

[tex]\displaystyle s_n=342[/tex]

[tex]\displaystyle a_n=3a_1[/tex]

Using the last condition in eq 1 and knowing that r=1 (consecutive numbers)

[tex]\displaystyle a_n=a_1+n-1=3a_1[/tex]

Rearranging

[tex]\displaystyle 2a_1=n-1[/tex]

Using eq 2

[tex]\displaystyle \frac{a_1+a_n}{2}n=342[/tex]

Replacing the first condition

[tex]\displaystyle \frac{a_1+3a_1}{2}n=342[/tex]

Simplifying

[tex]\displaystyle 2a_1\ n=342[/tex]

Since  

[tex]\displaystyle 2a_1=n-1[/tex]

We have

[tex]\displaystyle n(n-1)=342[/tex]

Factoring

[tex]\displaystyle n(n-1)=(19)(18)[/tex]

We find the number of terms

[tex]\displaystyle n=19[/tex]

The first term is

[tex]\displaystyle a_1=\ \frac{342}{38}=9[/tex]

Final answer:

The smallest integer is 6, and the sequence contains 19 terms.

Explanation:

To solve the problem about a sequence of consecutive integers where the sum is 342 and the largest integer is three times the smallest integer, we will use the formula for the sum of an arithmetic sequence and set up a system of equations. The sum of an arithmetic sequence is given by: S = ½ n(first integer + last integer), where S is the sum of the sequence, n is the number of terms, the first integer is a, and the last integer is l. We are given S = 342 and l = 3a.

Let's set up the system of equations:

By substituting l = 3a into the first equation, we get:

Hence, n and a must be factors of 684 (since 342 = 2 × 171 = 4 × 342). Through trial and error or using a system of linear equations, we can find the appropriate values of n and a that will satisfy both the sum and the relationship between the smallest and largest integers.

Ultimately, we find that the smallest integer in the sequence is 6, and the sequence contains 19 terms.

Answer:

Null hypothesis:[tex]p_{1} = p_{2}[/tex]  

Alternative hypothesis:[tex]p_{1} \neq p_{2}[/tex]

[tex]z=\frac{0.41-0.43}{\sqrt{0.42(1-0.42)(\frac{1}{1300}+\frac{1}{1300})}}=-1.033[/tex]    

[tex]p_v =2*P(Z<-1.033)=0.302[/tex]  

So the p value is a very high value and using any significance level for example [tex]\alpha=0.05, 0,1,0.15[/tex] always [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the the proportion 1 is not significantly different from the proportion 2.  

Step-by-step explanation:

1) Data given and notation  

n = 1300 sample size selected

[tex]p_{1}=0.41[/tex] represent the proportion of adults approve of President1.

[tex]p_{2}=0.42[/tex] represent the proportion of adults approve of President2.

z would represent the statistic (variable of interest)  

[tex]p_v[/tex] represent the value for the test (variable of interest)  

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to check if the proportion 1 is different from proportion 2  , the system of hypothesis would be:  

Null hypothesis:[tex]p_{1} = p_{2}[/tex]  

Alternative hypothesis:[tex]p_{1} \neq p_{2}[/tex]  

We need to apply a z test to compare proportions, and the statistic is given by:  

[tex]z=\frac{p_{1}-p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}[/tex]   (1)  

Where [tex]\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{0.41+0.43}{2}=0.42[/tex]  

3) Calculate the statistic  

Replacing in formula (1) the values obtained we got this:  

[tex]z=\frac{0.41-0.43}{\sqrt{0.42(1-0.42)(\frac{1}{1300}+\frac{1}{1300})}}=-1.033[/tex]    

4) Statistical decision

For this case we don't have a significance level provided [tex]\alpha[/tex], but we can calculate the p value for this test.    

Since is a two sided test the p value would be:  

[tex]p_v =2*P(Z<-1.033)=0.302[/tex]  

So the p value is a very high value and using any significance level for example [tex]\alpha=0.05, 0,1,0.15[/tex] always [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the the proportion 1 is not significantly different from the proportion 2.  

Answer:

[tex] \frac{1}{5} \: and \: \frac{1}{15} [/tex]

Both of these must have similar denominators:

[tex] \frac{3}{15} \: and \: \frac{1}{15} \\ \frac{3}{15} \: i s \: \: more \: than \: \frac{1}{15} \\ \frac{3}{15} > \frac{1}{15} [/tex]

Good luck!

Intelligent Muslim,

From Uzbekistan.

The integral of [tex]z[/tex] over [tex]S[/tex] is equal to the sum of the integrals of [tex]z[/tex] over [tex]S_1,S_2,S_3[/tex].

Parameterize the surface by

[tex]\vec r(u,v)=(9\cos u,9\sin u,v)[/tex]

with [tex]0\le u\le2\pi[/tex] and [tex]0\le v\le9+9\cos u[/tex]. Take the normal vector to [tex]S_1[/tex] to be

[tex]\dfrac{\partial\vec r}{\partial u}\times\dfrac{\partial\vec r}{\partial v}=(9\cos u,9\sin u,0)[/tex]

Then the integral of [tex]z[/tex] over [tex]S_1[/tex] is

[tex]\displaystyle\iint_{S_1}z\,\mathrm dS=\int_0^{2\pi}\int_0^{9+9\cos u}v\|(9\cos u,9\sin u,0)\|\,\mathrm du\,\mathrm dv=\frac{2187\pi}2[/tex]

Parameterize [tex]S_2[/tex] by

[tex]\vec s(u,v)=(u\cos v,u\sin v,0)[/tex]

with [tex]0\le u\le9[/tex] and [tex]0\le v\le2\pi[/tex]. Since [tex]z=0[/tex], the integral over [tex]S_2[/tex] is also 0.

Parameterize [tex]S_3[/tex] by

[tex]\vec t(u,v)=(u\cos v,u\sin v,9+u\cos v)[/tex]

with [tex]0\le u\le9[/tex] and [tex]0\le v\le2\pi[/tex]. The normal to [tex]S_3[/tex] is

[tex]\dfrac{\partial\vec t}{\partial u}\times\dfrac{\partial\vec t}{\partial v}=(-u,0,u)[/tex]

so that the integral over [tex]S_3[/tex] is

[tex]\displaystyle\iint_{S_3}z\,\mathrm dS=\int_0^{2\pi}\int_0^9(9+u\cos v)\|(-u,0,u)\|\,\mathrm du\,\mathrm dv=729\sqrt2\,\pi[/tex]

Putting the results together, the integral of [tex]z[/tex] over [tex]S[/tex] is

[tex]\iint_Sz\,\mathrm dS=\boxed{\left(\frac{2187}2+729\sqrt2\right)\pi}[/tex]

The procedure to evaluate this integral begins by evaluating integrals over the three distinct parts of the surface: the side, bottom, and top. Both the side and bottom integrals are zero and the integral over the top can be evaluated using polar coordinates, however, without specified boundaries for the upper bound, we cannot determine the exact result.

The task is to evaluate a surface integral over a surface S, which consists of a cylinder, a disk on the bottom, and a plane on the top. The surface integral of a scalar function z over a surface S is computed by taking the antiderivative of the function with respect to both dimensions defining the area. Since S consists of three parts, S1 (the side), S2 (the bottom), and S3 (the top), we must perform this process for each part and sum the results.

The key steps in the procedure are:

Summing the results of our calculations on S1, S2, and S3 would yield the final answer. Unfortunately, without the boundaries for the upper bound for the integration, we cannot provide the exact result.

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Answer: She should blend 98 lbs of high-quality beans.

She should blend 72 lbs of cheaper beans

Step-by-step explanation:

Let x represent the number of pounds of high quality beans that she should blend.

Let y represent the number of pounds of cheaper beans that she should blend.

She needs to prepare 170 lbs of blended coffee beans. This means that

x + y = 170

She plans to do this by blending together a high-quality bean costing $4.75 per pound and a cheaper bean at $2.00 per pound. The blend would sell for $3.59 per pound. This means that the total cost of the blend would be 3.59×170 = $610.3. This means that

4.75x + 2y = 610.3 - - - - - - - - - -1

Substituting x = 170 - y into equation 1, it becomes

4.75(170 - y) + 2y = 610.3

807.5 - 4.75y + 2y = 610.3

- 4.75y + 2y = 610.3 - 807.5

- 2.75y = - 197.2

y = - 197.2/-2.75 = 71.9

y = 72 pounds

x = 170 - y = 170 - 71.9

x = 98.1

x = 98 pounds