A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time is less than 7.6 hours.

Answer: Simple random sampling

Step-by-step explanation:

Simple random sampling is a technique of sampling in which an experimenter selects the group of samples or subjects randomly from a large group of population. Each sample is chosen randomly by chance and each entity of a population has equal possibility of being selected as sample.

According to the given situation, simple random sampling is the technique that should be used for sampling.

The equation of the line indicates that the x and y-intercept and the formula for the area of the triangle are;

x-intercept (c, 0)

y-intercept (0, 1/(c² + 7))

A(c) = c/(2·(c² + 7))

The steps used to find the x-intercept and the y-intercept are presented as follows;

The equation of the line c·(c² + 7)·y = c - x, can be expressed in the slope intercept form to find the coordinates of the x-intercept and the coordinates of the y-intercept as follows;

c·(c² + 7)·y = c - x

y = (c - x)/(c·(c² + 7))

y = c/(c·(c² + 7)) - x/(c·(c² + 7))

y = 1/((c² + 7)) - x/(c·(c² + 7))

The above equation is in the slope-intercept form, y = m·x + c

Where c is the y-coordinate of the y-intercept, and (0, c) ids the coordinate of the y-intercept; Therefore, the coordinates of the y-intercept is; (0,  1/((c² + 7)))

The coordinate of the x-intercept can be found by plugging in y = 0, in the above equation to get;

0 = 1/((c² + 7)) - x/(c·(c² + 7))

x/(c·(c² + 7)) = 1/((c² + 7))

x = (c·(c² + 7))/((c² + 7))

(c·(c² + 7))/((c² + 7)) = c

x = c

Therefore coordinates of the x-intercept is; (c, 0)

The triangle enclosed by the line and the x-axis and y-axis is a right triangle, therefore;

The positive x and y-values of the x-intercept and y-intercept indicates that the area of the triangle is the product half the distance from the origin to the y-intercept and the distance from the origin to the x-intercept

Area = (1/2) × (1/((c² + 7)) - 0) × (c - 0)

(1/2) × (1/((c² + 7))) × (c) = c/(2·(c² + 7)

Area of the triangle, A(c) = c/(2·(c² + 7)

The complete question found through search can be presented as follows;

Consider the equation of the line c·(c² + 7)·y = c - x where c > 0 is a constant

(a) Find the coordinates of the x-intercept and the y-intercept

x-intercept (  ,  )

y-intercept (  ,  )

(b) Find a formula for the area of the triangle enclosed between the line, the x-axis and the y-axis

Answer:

  48°

Step-by-step explanation:

In triangle CYZ, angles Y and C are complementary, so ...

  (5x +22) + (12x) = 90

  17x +22 = 90

  17x = 68

  x = 4

The angle of interest is 12x° , so has measure

  12·4° = 48°

Answer:

m∠WZC=48°

Step-by-step explanation:

we know that

If ZC is an altitude

then

Triangle YZC is a right triangle

therefore

m∠CYW+m∠WZC=90° ----> by complementary angles

substitute

(5x+22)°+(12x)°=90°

17x=(90-22)°

x=4°

Find the measure of angle WZC

m∠WZC=(12x)°=12*4=48°

Final answer:

Train B travels at a speed of 30 miles per hour and train A travels at a speed of 50 miles per hour.

Explanation:

Let's say the speed of train B is x miles per hour. Since train A is 20 miles per hour faster, the speed of train A is x + 20 miles per hour.

Distance = Speed x Time

For train A, Distance = (x +20) * 3

For train B, Distance = x * 3

Since they meet 240 miles apart, the sum of their distances is 240:

(x + 20) * 3 + x * 3 = 240

3x + 60 + 3x = 240

6x = 180

x = 30

Hence, train B travels at a speed of 30 miles per hour and train A travels at a speed of 50 miles per hour.

Answer:

r^8

Step-by-step explanation:

r 2 • r • r 5

Assuming that r 2 means r^2

r ^2 • r • r^ 5

When the bases are the same and we are multiplying, we add the exponents

r ^2 • r^1 • r^ 5

r^(2+1+5)

r^(8)

Answer: [tex]r^8[/tex]

Step-by-step explanation:

You need to remember a property called "Product of powers property". This property states the following:

[tex](a^m)(a^n)=a^{(m+n)}[/tex]

You can observe that the bases of the expression [tex]r^2*r*r^5[/tex] are equal, then you can apply the property mentioned before.

Knowing this, you can make the multiplication indicated. Therefore, you get that the product is:

[tex]r^2*r*r^5=r^{(2+1+5)}=r^8[/tex]

Answer:

(3;4); (-3;-4); (3;-4); (-3;4)

Step-by-step explanation:

for more information see the attached picture.

Answer:  The required solution set is

(x, y) = (3, 4), (-3, 4), (3, -4) and (-3, -4).

Step-by-step explanation:  We are given to solve the following system :

[tex]x^2+y^2=25~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\y^2-x^2=7~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]

We will be using the method of Elimination to solve the problem.

Adding equations (i) and (ii), we have

[tex](x^2+y^2)+(y^2-x^2)=25+7\\\\\Rightarrow 2y^2=32\\\\\Rightarrow y^2=16\\\\\Rightarrow y=\pm\sqrt{16}~~~~~~~~~~~~~~~~~~~[\textup{taking square root on both sides}]\\\\\Rightarrow y=\pm4.[/tex]

From equation (ii), we get

[tex](\pm4)^2-x^2=7\\\\\Rightarrow 16-x^2=7\\\\\Rightarrow x^2=16-7\\\\\Rightarrow x^2=9\\\\\Rightarrow x=\pm\sqrt9~~~~~~~~~~~~~~~~~~[\textup{taking square root on both sides}]\\\\\Rightarrow x=\pm3.[/tex]

Thus, the required solution set is

(x, y) = (3, 4), (-3, 4), (3, -4) and (-3, -4).

Answer:

[tex]26.12\:<\:\mu\:<\:31.48[/tex]

Step-by-step explanation:

Since the population standard deviation [tex]\sigma[/tex] is unknown, and the sample standard deviation [tex]s[/tex], must replace it, the [tex]t[/tex] distribution  must be used for the confidence interval.

The sample size is n=8.

The degree of freedom is [tex]df=n-1[/tex], [tex]\implies df=8-1=7[/tex].

With 95% confidence level, the [tex]\alpha-level[/tex](significance level) is 5%.

Hence with 7 degrees of freedom, [tex]t_{\frac{\alpha}{2} }=2.365[/tex]. (Read from the t-distribution table see attachment)

The 95% confidence interval can be found by using the formula:

[tex]\bar X-t_{\frac{\alpha}{2}}(\frac{s}{\sqrt{n} } )\:<\:\mu\:<\:\bar X+t_{\frac{\alpha}{2}}(\frac{s}{\sqrt{n} } )[/tex].

The sample mean is [tex]\bar X=28.8[/tex] hours.

The sample sample standard deviation is [tex]s=3.2[/tex] hours.

We now substitute all these values into the formula to obtain:

[tex]28.8-2.365(\frac{3.2}{\sqrt{8} } )\:<\:\mu\:<\:28.8+2.365(\frac{3.2}{\sqrt{8} } )[/tex].

[tex]26.12\:<\:\mu\:<\:31.48[/tex]

We are 95% confident that the population mean is between 26.12 and 31.48  hours.

Answer: The altitude and the base of the sign are 6 meters and 11 meters respectively.

Step-by-step explanation:

Since we have given that

Area of triangular sign = 33 sq. meters

Let the altitude of the triangle be 'x'.

Let the base of the triangle be ' 2x-1'.

As we know the formula for "Area of triangle ":

[tex]Area=\dfrac{1}{2}\times base\times height\\\\33=\dfrac{1}{2}\times x(2x-1)\\\\33\times 2=2x^2-x\\\\66=2x^2-x\\\\2x^2-x-66=0\\\\2x^2-12x+11x-66=0\\\\2x(x-6)+11(x-6)=0\\\\(2x+11)(x-6)=0\\\\x=-\dfrac{11}{2},6\\\\x=-5.5,6[/tex]

Discarded the negative value of x for dimensions:

So, altitude of triangle becomes 6 meters

Base of triangle would be [tex]2(6)-1=12-1=11\ meters[/tex]

For this case we must factor the following expression:

[tex]x ^ 3-y ^ 3[/tex]

We have that since both terms are perfect cubes, we factor using the cube difference formula.

[tex]a ^ 3-b ^ 3 = (a-b) (a ^ 2 + ab + b ^ 2)[/tex]

Where:

[tex]a = x\\b = y[/tex]

So, we have to:

[tex]x ^ 3-y ^ 3 = (x-y) (x ^ 2 + xy + y ^ 2)[/tex]

Answer:

[tex]x ^ 3-y ^ 3 = (x-y) (x ^ 2 + xy + y ^ 2)[/tex]

Answer:

Company's total liabilities are $6,351.

Step-by-step explanation:

A company's financial information is given, we have to calculate the company's total liabilities.

Total assets = Fixed assets + current assets

Current assets = Total assets - Fixed assets

                         = $8,190 - $5,385

                         = $2,805

Networking capital = Current assets - Current liabilities

                        $735 = $2,805 - current liabilities

  Current liabilities   = $2,805 - $735

  Current liabilities   = $2,070

Total liabilities = Long term debt + current liabilities

                         = $4,281 + $2,070

                         = $6,351

Company's total liabilities are $6,351.

Answer:

Equation of tangent is [tex]y-2\sqrt{3}x=2-2\frac{\pi}{\sqrt{3}}[/tex]

Step-by-step explanation:

Given:

Equation of curve, y = sec x

Passing through point = [tex](\frac{\pi}{3},2)[/tex]

We need to find Equation of tangent to the given curve and at the given point.

First we find the slope of the tangent by differentiating the given curve.

As we know that slope of the tangent, m = [tex]\frac{\mathrm{d}y}{\mathrm{d} x}[/tex]

So, consider

y = sec x

[tex]\frac{\mathrm{d}y}{\mathrm{d} x}=\frac{\mathrm{d}(sec\,x)}{\mathrm{d} x}[/tex]

[tex]\frac{\mathrm{d}y}{\mathrm{d} x}=sec\,x\:tan\,x[/tex]

Now we find value of slope at given point,

put x = [tex]\frac{\pi}{3}[/tex] in above derivate

we get

[tex]\frac{\mathrm{d}y}{\mathrm{d} x}=sec\,\frac{\pi}{3}\:tan\,{\pi}{3}=2\sqrt{3}[/tex]

Now using Slope point form, we have

Equation of tangent

[tex]y-y_1=m(x-x_1)[/tex]

[tex]y-2=2\sqrt{3}(x-\frac{\pi}{3})[/tex]

[tex]y=2\sqrt{3}x-2\sqrt{3}\frac{\pi}{3}+2[/tex]

[tex]y-2\sqrt{3}x=2-2\frac{\pi}{\sqrt{3}}[/tex]

Therefore, Equation of tangent is [tex]y-2\sqrt{3}x=2-2\frac{\pi}{\sqrt{3}}[/tex]

Answer:

There are 13 families had a parakeet only

Step-by-step explanation:

* Lets explain the problem

- There are 180 families

- 67 families had a dog

- 52 families had a cat

- 22 families had a dog and a cat

- 70 had neither a cat nor a​ dog, and in addition did not have a​

 parakeet

- 4 had a​ cat, a​ dog, and a parakeet (4 is a part of 22 and 22 is a part

 of 67 and 520

* We will explain the Venn-diagram

- A rectangle represent the total of the families

- Three intersected circles:

 C represented the cat

 D represented the dog

 P represented the parakeet

- The common part of the three circle had 4 families

- The common part between the circle of the cat and the circle of the

 dog only had 22 - 4 = 18 families

- The common part between the circle of the dog and the circle of the

 parakeet only had a families

- The common part between the circle of the cat and the circle of the

 parakeet only had b families

- The non-intersected part of the circle of the dog had 67 - 22 - a =

  45 - a families

  had dogs only

- The non-intersected part of the circle of the cat had 52 - 22 - b =  

  30 - b families

  had cats only

- The non-intersected part of the circle of the parakeet had c families

  had parakeets only

- The part out side the circles and inside the triangle has 70 families

- Look to the attached graph for more under stand

∵ The total of the families is 180

∴ The sum of all steps above is 180

∴ 45 - a + 18 + 4 + 30 - b + b + c + a + 70 = 180 ⇒ simplify

- (-a) will cancel (a) and (-b) will cancel (b)

∴ (45 + 18 + 4 + 30 + 70) + (-a + a) + (-b + b) + c = 180

∴ 167 + c = 180 ⇒ subtract 167 from both sides

∴ c = 180 - 167 = 13 families

* There are 13 families had a parakeet only

Answer: 19.77%

Step-by-step explanation:

Given: Mean : [tex]\mu=74[/tex]

Standard deviation : [tex]\sigma = 7.1[/tex]

The formula to calculate z-score is given by :_

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

For x= 80, we have

[tex]z=\dfrac{80-74}{7.1}\approx0.85[/tex]

The P-value = [tex]P(z>0.85)=1-P(z<0.85)=1-0.8023374=0.1976626[/tex]

In percent , [tex]0.1976626\times100=19.76626\%\approx19.77\%[/tex]

Hence, the approximate percentage of the test scores during the past year exceeded 80 =19.77%

The probability that the customer did not try on the t- shirt is:

                             0.2432

It is given that the probability that a customer tries a t-shirt is: 0.40

After he will try

The probability that he will buy the t-shirt is: 0.70

The probability that he will not try t-shirt is: 1-0.40=0.60

If he will not try

then the probability of buying a t-shirt is: 0.15

This means that the total probability of Buying a t-shirt is:

  0.40×0.70+0.60×0.15

= 0.37

It is given that the customer bought a t-shirt, we need to find the probability that he did not try the  t-shirt:

The probability is given by:

[tex]\dfrac{0.60\times 0.15}{0.37}\\\\\\=\dfrac{0.09}{0.37}\\\\\\=0.2432[/tex]

The question asks for the conditional probability that a customer did not try on a t-shirt given that they bought it. This probability value can be calculated using the total probability theorem and Bayes' theorem. It is found to be approximately 0.159.

This question falls under the topic of probability, specifically conditional probability and Bayes' theorem. The probability values are given in the problem statement but it seeks the probability that a customer did not try on a t-shirt given that they bought it.

Let A represent the event that a customer buys a t-shirt and B1 represent the event that a customer tries on a t-shirt whereas B2 represent the event that the customer did not try on a t-shirt.

We are given that P(B1) = 0.40, P(A|B1) = 0.70 and P(A|B2) = 0.15. We are asked to find P(B2|A).

To find P(B2|A), we use Bayes' theorem:

P(B2|A) = (P(B2) * P(A|B2)) / P(A)

We know that P(B1) + P(B2) = 1, so P(B2) = 1 - P(B1) = 1 - 0.40 = 0.60.

The probability that a customer tried a t-shirt is 0.40 and bought it is 0.70, that yields the joint probability P(A, B1) = P(B1) * P(A|B1) = 0.40 * 0.70 = 0.28. Similarly the joint probability P(A, B2) = P(B2) * P(A|B2) = 0.60 * 0.15 = 0.09.

P(A) is the probability that a customer buys a t-shirt and is calculated using the total probability theorem as P(A) = P(A, B1) + P(A, B2) = 0.28 + 0.09 = 0.37.

Substituting the values we calculated into Bayes' theorem, we find:

P(B2|A) = (0.60 * 0.15) / 0.37 ≈ 0.24.

So, the probability that a customer did not try the t-shirt given that they bought it is approximately 0.24.

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Answer:

Overall Final Score = 83.7

Letter Grade = B

Step-by-step explanation:

To calculate weighted marks (grade), the rule is to multiply each weight  (in decimal) with respective marks and add them together. Let's do this:

Overall final score = (0.10*74) + (0.25*93) + (0.50*83) + (0.15*77) = 83.7

From the information given, we know that score between 80 and 90 is a "B".

Answer: c) The number 75 is an outlier on the stem and leaf plot.

Step-by-step explanation:

When we look in the given stem-leaf plot, there is no leaf attached to the stem with value 6.

It mean there is no value between 50 and 75.

It shows that the value of 75 is detached from the other values in the data.

⇒ The number 75 is an outlier on the stem and leaf plot.

Answer:

The answer is C

Step-by-step explanation:

Answer:

543.7

Step-by-step explanation:

Answer:

543.7, 543.6, 543.7, 543.7, 543., 544.0

Step-by-step explanation:

Answer:

4th Option is correct.

Step-by-step explanation:

Given:

Number of apples in refrigerator = 6

Number of oranges in refrigerator = 5

Number of bananas in refrigerator = 10

Number of pears in refrigerator = 3

Number of peaches in refrigerator = 7

Number of plums in refrigerator = 11

Number of mangoes in refrigerator = 2

A piece is randomly taken out from refrigerator.

To find: Sample Space of the experiment

Sample Space : It is a set which contain all the possible outcome / results of the experiment.

So, here Sample Space = { 6 apples , 5 oranges , 10 bananas , 3 pears , 7 peaches , 11 plums , 2 mangoes }

Therefore, 4th Option is correct.

The line y = 3 means that x = 0.

The point (-3, y) tells me that x = -3 when y is 3.

So, y = 3 completes the point (-3, 3).

Try this option:

the functions y=sin(x) and y=cos(x) are the standart functions; it means, the period of each is 2π, maximum value is '1', minimum value is '-1'.

Using the properties described above, to find the max. value for the function y=2cos(x): 2*1=2, where 2- the amplitude of the given function, 1- maximum of the standart function;

the max. value for the function y=3sin(x+π): 3*1=3, where 3 - the amplitude of the given function, 1- maximum of the standart function.

For more details see the attached picture.

In other words, common equation for such functions is y=A*sin(ωx+Ф), where A- amplitude, ω - frequency and Ф - initial phase of the given function.

Max. and min. values depend on A, all the points of the function repeat every 2π, including max. and min. values.

[tex]f(x)=2\cos x[/tex]

We know that the function f(x) is a cosine function and the maximum value is obtained by the function when this cosine function takes the maximum value.

We know that:

[tex]-1\leq \cos x\leq 1[/tex]

This means that the cosine function  takes the maximum values as: 1

and when [tex]\cos x=1[/tex] then

[tex]f(x)=2\times 1\\\\i.e.\\\\f(x)=2[/tex]

i.e. Maximum value of function f(x)=2

[tex]g(x)=3\sin (x+\pi)[/tex]

Again the function g(x) will attain the maximum value when the function sine will takes the maximum value.

We know that the range of the sine function is: [-1,1]

This means that the maximum value of sine function is: 1

Hence, when [tex]\sin (x+\pi)=1[/tex] then,

[tex]g(x)=3\times 1\\\\i.e.\\\\g(x)=3[/tex]

i.e. Maximum value of function g(x)=3

          The function g(x) has the largest maximum value.

                            ( Since 3>2)

Answer with explanation:

Given : Sample mean =[tex]\overline{x}=\text{48.47 pounds}[/tex]

Standard deviation : [tex]\sigma=\text{ 3.1 pounds.}[/tex]

Sample size : n = 36

Claim : [tex]\mu\neq50[/tex]

∴ [tex]H_0:\mu=50[/tex]

[tex]H_1:\mu\neq50[/tex]

Since the alternative hypothesis is two tail , then the test is two tail test.

By using a z statistic and a 0.05 level of significance. Reject  [tex]H_0[/tex] if z < -1.960 or is z> 1.960.

Then , the test static for population mean is given by :-

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

[tex]z=\dfrac{48.47-50}{\dfrac{3.1}{\sqrt{36}}}\approx-2.96[/tex]

We reject [tex]H_0[/tex] since [tex]-2.96\leq -1.96[/tex]. We have statistically significant evidence at [tex]\alpha=0.05[/tex] to show that the mean amount of garbage per bin is not different from 50.  

To test if the mean amount of garbage per bin is different from 50, we set up null and alternative hypotheses and use a z-test to calculate a z-score. The z-score of -1.49 falls within the acceptance region; hence, we fail to reject the null hypothesis, implying that the mean amount of garbage per bin is not significantly different from 50 pounds.

First, we need to set up our null and alternative hypotheses. The null hypothesis (H0) states that the mean amount of garbage is 50, or μ = 50. The alternative hypothesis (H1) is that the mean is different from 50, or μ ≠ 50.

Second, calculate the z-score which is given by the formula Z = (X - μ) / (σ/√n), where X is the sample mean, μ is the population mean, σ is the standard deviation and n is the sample size. Substituting the given values, we get Z = (48.47 - 50) / (3.1 / √36) = -1.49

For a two-tailed test, we are checking if the z-score falls within the rejection region. Since our level of significance is 0.05, the critical values for a two-tailed test are -1.96 and 1.96. Our computed z-score of -1.49 falls within the acceptance region. Therefore, we fail to reject the null hypothesis. This suggests that the mean amount of garbage per bin is not statistically different from 50 pounds at the 0.05 level of significance.

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Answer:

c

Step-by-step explanation:

Answer:

56.25%

Step-by-step explanation:

If 14 are male, that means that 32 - 14 = 18 are female.  If 18/32 are female, do the division and then multiply by 100 to get the percentage.

18/32 = .5625

.5625 × 100% = 56.25%

Answer: 0.6915

Step-by-step explanation:

Given : [tex]\text{Mean}=\mu=10^{\circ}C[/tex]

[tex]\text{Standard deviation}=\sigma=10^{\circ}C[/tex]

Since , the distribution follows a Normal distribution.

The formula to calculate the z-score is given by :-

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

For x=[tex]15^{\circ}C[/tex]

[tex]z=\dfrac{15-10}{10}=0.5[/tex]

The p-value = [tex]P(z\leq0.5)=0.6914625\approx0.6915[/tex]

Hence, the required probability : 0.6915

The probability that the temperature will be less than or equal to 15 degrees Celsius in a city with a mean temperature of 10 degrees Celsius and a standard deviation of 10 degrees Celsius is approximately 69.15%.

To find the probability that the temperature will be less than or equal to 15 degrees Celsius when the mean and standard deviation are both 10 degrees Celsius, we can calculate the z-score.

The z-score formula is:

z = (X - μ) / Σ

For X = 15, μ = 10, and Σ = 10, the z-score is:

z = (15 - 10) / 10 = 0.5

We then look up the z-score of 0.5 in a standard normal distribution table or use a calculator to find the cumulative probability corresponding to the z-score. The probability that the temperature is less than or equal to 15 degrees Celsius is approximately 0.6915 (or 69.15%).

Final answer:

To achieve a 2:3 ratio for the picture frame's dimensions, 6 inches must be cut from both the length and width.

Explanation:

To resize the picture frame while maintaining a specific ratio between its sides, we need to establish a proportion based on the desired ratio of 2:3 (shorter side to the longer side). Let's denote the amount that needs to be cut from the shorter side as x inches, and the amount to be cut from the longer side as y inches. After cutting, the dimensions of the frame will be (33 - y) inches by (24 - x) inches.

We want to set up a proportion to reflect the desired ratio: (24 - x)/(33 - y) = 2/3. To solve for x and y, we can use the property that cross-multiplying the terms in a proportion gives us an equality: 3(24 - x) = 2(33 - y). This simplifies to 72 - 3x = 66 - 2y. We also know that the amount cut off from both dimensions should be the same, thus x = y. Substituting y for x gives us 72 - 3x = 66 - 2x. Solving this equation, we get x = 6 inches. So, 6 inches must be cut from both the length and the width to achieve the 2:3 ratio.

Answer:

A

Step-by-step explanation:

Sum of the first n terms of a geometric series is:

S = a₁ (1 - r^n) / (1 - r)

Here, a₁ = 0.2, r = 6, and n = 5.

S = 0.2 (1 - 6^5) / (1 - 6)

S = 311

Answer:

Option A

Step-by-step explanation:

For the given geometric series  

a₁ = 0.2

a₅ = 259.2

r = 6

Then we have to find the sum of initial 5 terms of this series

[tex]S_{n} =\frac{a_1(r^4-1)}{(r-1)}=\frac{0.2(6^3-1)}{(6-1)}[/tex]

[tex]=\frac{0.2(7776-1)}{5}[/tex]

[tex]\frac{0.2\times 7775}{5}[/tex]

= 311

Option A is the answer.

Answer:

The mass of the lamina is 1

Step-by-step explanation:

Let [tex]\rho(x,y)[/tex] be a continuous density function of a lamina in the plane region D,then the mass of the lamina is given by:

[tex]m=\int\limits \int\limits_D \rho(x,y) \, dA[/tex].

From the question, the given density function is [tex]\rho (x,y)=xye^{x+y}[/tex].

Again, the lamina occupies a rectangular region: D={(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}.

The mass of the lamina can be found by evaluating the double integral:

[tex]I=\int\limits^1_0\int\limits^1_0xye^{x+y}dydx[/tex].

Since D is a rectangular region, we can apply Fubini's Theorem to get:

[tex]I=\int\limits^1_0(\int\limits^1_0xye^{x+y}dy)dx[/tex].

Let the inner integral be: [tex]I_0=\int\limits^1_0xye^{x+y}dy[/tex], then

[tex]I=\int\limits^1_0(I_0)dx[/tex].

The inner integral is evaluated using integration by parts.

Let [tex]u=xy[/tex], the partial derivative of u wrt y is

[tex]\implies du=xdy[/tex]

and

[tex]dv=\int\limits e^{x+y} dy[/tex], integrating wrt y, we obtain

[tex]v=\int\limits e^{x+y}[/tex]

Recall the integration by parts formula:[tex]\int\limits udv=uv- \int\limits vdu[/tex]

This implies that:

[tex]\int\limits xye^{x+y}dy=xye^{x+y}-\int\limits e^{x+y}\cdot xdy[/tex]

[tex]\int\limits xye^{x+y}dy=xye^{x+y}-xe^{x+y}[/tex]

[tex]I_0=\int\limits^1_0 xye^{x+y}dy[/tex]

We substitute the limits of integration and evaluate to get:

[tex]I_0=xe^x[/tex]

This implies that:

[tex]I=\int\limits^1_0(xe^x)dx[/tex].

Or

[tex]I=\int\limits^1_0xe^xdx[/tex].

We again apply integration by parts formula to get:

[tex]\int\limits xe^xdx=e^x(x-1)[/tex].

[tex]I=\int\limits^1_0xe^xdx=e^1(1-1)-e^0(0-1)[/tex].

[tex]I=\int\limits^1_0xe^xdx=0-1(0-1)[/tex].

[tex]I=\int\limits^1_0xe^xdx=0-1(-1)=1[/tex].

No unit is given, therefore the mass of the lamina is 1.

The mass of the lamina is (1 − e)/2 units.

To find the mass of the lamina occupying the region D = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} with the density function ρ(x, y) = xy[tex]e^{(x+y)[/tex] , we need to calculate the double integral of the density function over the given region.

M = ∬D ρ(x,y) dA

Since D is a rectangular region, we can write:

M = ∫01 ∫01 xy [tex]e^{(x+y)[/tex] dy dx

∫01 xy [tex]e^{(x+y)[/tex] dy = x ∫01 y [tex]e^{(x+y)[/tex] dy

Using integration by parts where u = y, dv = [tex]e^{(x+y)[/tex] dy:

∫ y [tex]e^{(x+y)[/tex] dy = [y [tex]e^{(x+y)[/tex] / ex] 01 - ∫ [tex]e^{(x+y)[/tex] / ex dy

= [y [tex]e^{(x+y)[/tex]] 01 - [[tex]e^{(x+y)[/tex]] 01

= [[tex]e^{(x+1)[/tex]) - eˣ - [tex]e^{(x+1)[/tex] + e⁰]/eˣ

= x (1 − e¹)

∫01 x (1 − e¹) dx = (1 − e¹) ∫01 x dx

= (1 − e¹) [x2 / 2] 01

= (1 − e¹)(1/2)

= (1 − e¹)/2

The mass of the lamina is (1 − e)/2 units.

Answer: 70°

Step-by-step explanation:

ΔVWX ≅ ΔNOP     ⇒    ∠V=∠N, ∠W=∠O, and ∠X=∠P

Since ∠V=44° and ∠P=66°=∠X

then we can use the Triangle Sum Theorem:

∠V + ∠W + ∠X = 180°

44° + ∠W + 66° = 180°

        ∠W + 110° = 180°

        ∠W           = 70°

I'm going to assume this reads

[tex]\vec F(x,y,z)=5e^x\sin y\,\vec\imath+5e^x\cos y\,\vec\jmath+7z^2\,\vec k[/tex]

and the path [tex]C[/tex] is parameterized by

[tex]\vec c(t)=8t\,\vec\imath+t^3\,\vec\jmath+e^t\,\vec k[/tex]

with [tex]0\le t\le1[/tex]. Under this parameterization,

[tex]\vec F(x,y,z)=\vec F(x(t),y(t),z(t))=5e^{8t}\sin(t^3)\,\vec\imath+5e^{8t}\cos(t^3)\,\vec\jmath+7e^{2t}\,\vec k[/tex]

and

[tex]\mathrm d\vec c=\dfrac{\mathrm d\vec c}{\mathrm dt}\,\mathrm dt=(8\,\vec\imath+3t^2\,\vec\jmath+e^t\,\vec k)\,\mathrm dt[/tex]

Then in the integral,

[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec s=\int_0^1\vec F(x(t),y(t),z(t))\cdot\frac{\mathrm d\vec c}{\mathrm dt}\,\mathrm dt[/tex]

[tex]=\displaystyle\int_0^1(7e^{3t}+15e^{8t}t^2\cos(t^3)+40e^{8t}\sin(t^3))\,\mathrm dt\approx\boxed{12586.5}[/tex]

(It's unlikely that an exact answer can be found in terms of elementary functions)