Find the area under the curve y =f( x) on [a,b] given f(x)=tan(3x) where a=0 b=pi/12

Answers

Answer 1

Answer:

The area under the curve y=f(x) on [a,b] is [tex]\frac{1}{6}\ln(2)[/tex] square units.

Step-by-step explanation:

The given function is

[tex]f(x)=\tan(3x)[/tex]

where a=0 and b=pi/12.

The area under the curve y=f(x) on [a,b] is defined as

[tex]Area=\int_{a}^{b}f(x)dx[/tex]

[tex]Area=\int_{0}^{\frac{\pi}{12}}\tan (3x)dx[/tex]

[tex]Area=\int_{0}^{\frac{\pi}{12}}\frac{\sin (3x)}{\cos (3x)}dx[/tex]

Substitute cos (3x)=t, so

[tex]-3\sin (3x)dx=dt[/tex]

[tex]\sin (3x)dx=-\frac{1}{3}dt[/tex]

[tex]a=\cos (3(0))=1[/tex]

[tex]b=\cos (3(\frac{\pi}{12}))=\frac{1}{\sqrt{2}}[/tex]

[tex]Area=-\frac{1}{3}\int_{1}^{\frac{1}{\sqrt{2}}}\frac{1}{t}dt[/tex]

[tex]Area=-\frac{1}{3}[\ln t]_{1}^{\frac{1}{\sqrt{2}}[/tex]

[tex]Area=-\frac{1}{3}(\ln \frac{1}{\sqrt{2}}-\ln (1))[/tex]

[tex]Area=-\frac{1}{3}(\ln 1-\ln \sqrt{2}-0)[/tex]

[tex]Area=-\frac{1}{3}(-\ln 2^{\frac{1}{2}})[/tex]

[tex]Area=-\frac{1}{3}(-\frac{1}{2}\ln 2)[/tex]

[tex]Area=-\frac{1}{6}\ln 2[/tex]

Therefore the area under the curve y=f(x) on [a,b] is [tex]\frac{1}{6}\ln(2)[/tex] square units.


Related Questions

x^3-y^3
Factor completely, then place the answer in the proper location on the grid.

Answers

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]

. Geometry The area of a triangular sign is 33 square meters. The base of the triangle is 1 meter less than double the altitude. Find the altitude and the base of the sign.

Answers

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]

Find an equation of the tangent line to the curve at the given point. y=secx, (pi/3, 2)

Answers

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]

In a sample of 8 high school students, they spent an average of 28.8 hours each week doing sports with a sample standard deviation of 3.2 hours. Find the 95% confidence interval, assuming the times are normally distributed.

Answers

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.

A sanitation supervisor is interested in testing to see if the mean amount of garbage per bin is different from 50. In a random sample of 36 bins, the sample mean amount was 48.47 pounds and the sample standard deviation was 3.1 pounds. Conduct the appropriate hypothesis test using a 0.05 level of significance.\

Answers

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.  

Final answer:

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.

Explanation:

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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An old picture frame has dimensions 33 inches by 24 inches. What one length must be cut from each dimension so that the ratio of the shorter side to the longer side is 2/3?

Answers

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.

What is the solution to the system of equations below when graphed y= -2x +3, y= -4x + 15

Answers

Answer:

(6, -9) → x = 6, y = -9

Step-by-step explanation:

These are linear functions. The graph is a straight line. We only need two points to draw a graph.

We choose any two values of x, substitute and calculate the value of y.

For y = -2x + 3

x = 0 → y =-2(0) + 3 = 0 + 3 = 3 → (0, 3)

x = 1 → y = -2(1) + 3 = - 2 + 3 = 1 → (1, 1)

For y = -4x + 15

x = 3 → y = -4(3) + 15 = -12 + 15 = 3 → (3, 3)

x = 4 → y = -4(4) + 15 = -16 + 15 = -1 → (4, -1)

Look at the picture.

The solution is the coordinates of the point of intersection of lines.

The solution to the system of equations below when graphed is (6, -9).

In order to graphically determine the solution for this system of linear equations on a coordinate plane, we would make use of an online graphing calculator to plot the given system of linear equations while taking note of the point of intersection.

When x is 0, 3, and 6, the y-value is given by;

y= -2(0) +3 = 3

y= -2(3) +3 = -3

y= -2(6) +3 = -9

y= -4(0) + 15 = 11

y= -4(3) + 15 = 3

y= -4(6) + 15 = -9

Since the y-intercept is at (0, 3) and the slope is 3, we would start with the point (0, 3), move 1 unit to the right and 3 units upward for the remaining points. For the second equation, the y-intercept is at (0, 15) and the slope is -4, we would start with the point (0, 15), move 1 unit to the right and 4 units downward for the remaining points.

Based on the graph shown, the solution set to the system is represented by the purple shaded region. Hence, a possible solution is (6, -9).

Functions f(x) and g(x) are shown below:

f(x) = 2cos(x)

g(x) = (3)sin(x+pi), a graph of sine function which starts at 0 comma 0 and decreases to the minimum of pi over 2 then increases to the maximum 3 pi over 2 then decreases to 2pi where the cycle repeats.

Using complete sentences, explain how to find the maximum value for each function and determine which function has the largest maximum y-value.

Please please please help!!!

Answers

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.

Answer with explanation:The function f(x) is given by:

[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

The function g(x) is given by:

[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)

Given the stem and leaf plot, which of the following statements is true?

STEM LEAF

2 9

3 2 6 7

4 1 2

5 0

6

7 5

a) There are no outliers on the stem and leaf plot; b) the numbers 29 and 75 are the outliers on the stem and leaf plot; c) the number 75 is an outlier on the stem and leaf plot; d) the number 60 is the outlier on the stem and leaf plot.

Answers

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

Step-by-step explanation:

An outlier is a value in data which is extremely large or small from all the values. It appears detached from the other values in data.

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:

Solve the following system. (Use (x,y) format in a single answer space.)
x2 + y2 = 25
y2 - x2 = 7
The solution set is

Answers

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).

A​ student's course grade is based on one midterm that counts as 10​% of his final​ grade, one class project that counts as 25​% of his final​ grade, a set of homework assignments that counts as 50​% of his final​ grade, and a final exam that counts as 15​% of his final grade. His midterm score is 74​, his project score is 93​, his homework score is 83​, and his final exam score is 77. What is his overall final​ score? What letter grade did he earn​ (A, B,​ C, D, or​ F)? Assume that a mean of 90 or above is an​ A, a mean of at least 80 but less than 90 is a​ B, and so on.

Answers

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".

A clothing store has just advertised at a bargain price, that a customer tries on a t-shirt is 0.40. a new range of t-shirts. The probability If a customer tries a t-shirt, the probability that they will buy it is 0.70 while if a customer does not try on a t-shirt, the probability that they will buy it is 0.15. A customer has just bought a t-shirt. Find the probability that the customer did not try on the t- shirt.

Answers

Answer:

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

                             0.2432

Step-by-step explanation:

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]

Final answer:

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.

Explanation:

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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Let F(x, y, z) = (5ex sin(y))i + (5ex cos(y))j + 7z2k. Evaluate the integral C F · ds, where c(t) = 8 t , t3, exp( t ) , 0 ≤ t ≤ 1. (Note that exp(u) = eu.)

Answers

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)

You find the following financial information about a company: net working capital = $735; fixed assets = $5,385; total assets = $8,190; and long-term debt = $4,281. What are the company's total liabilities?

Answers

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.

Find the mass of the lamina that occupies the region D = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} with the density function ρ(x, y) = xye^(x+y)

Answers

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.

Set up the double integral:

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

Evaluate the inner integral with respect to y first:

∫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:

du = dyv = [tex]e^{(x+y)[/tex] / eˣ

∫ 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)

Next, evaluate the outer integral:

∫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.

Round the following decimal to the nearest tenth.

543.67 ◯543.6 ◯543.68 ◯543.7 ◯543.670 ◯544​

Answers

Answer:

543.7

Step-by-step explanation:

Answer:

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

Step-by-step explanation:

In a classroom of 32 Students , 14 are male . What % of the Class is female ???? ​

Answers

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%

24 is divided into two parts such that 7 times the first part added to 5 times the second part makes 146. The first part is
a) 11 b) 13 c) 16 d ) 17

Answers

Answer:

13

Step-by-step explanation:

Let the two parts are x and y

as per the question

x+y=24   -----------------(A)

Also 7 times first part (x) that is 7x ,

when added to

5 times second part (y ) that is 5y

it gives 146

Hence our second equation becomes

7x+5y=146  ------------ (B)

Now we have to solve these two equations (A) and (B) to find the values of x and y

[tex]x+y=24 \\7x+5y=146\\[/tex]

now multiplying equation A with 5 and subtracting it from B

[tex]7x+5y=146[/tex]

[tex]-5x-5y=-120[/tex]

we get

2x=26

dividing both sides by 2 we get

x = 13

And hence our first part is 13

A city temperature is modeled as a normal random variable with mean and standard deviation both equal to 10 degrees Celsius. What is the probability that the temperature at a randomly chosen time will be less or equal to 15 degrees Celsius?

Answers

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%).

Find S for the given geometric series. Round answers to the nearest hundredth, if necessary.

a1=0.2, a5=259.2, r=6

a) 311
b)51
c)222.2
d)624.96

Answers

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.

A refrigerator contains 6 apples, 5 oranges, 10 bananas, 3 pears, 7 peaches, 11 plums, and 2 mangos. Imagine you stick your hand into the refrigerator and pull out a piece of fruit at random. What is the sample space for your action? ????={6, 5, 10, 3, 7, 11, 2}S={6, 5, 10, 3, 7, 11, 2} ????={apple, orange, banana, pear, peach, plum, mango}S={apple, orange, banana, pear, peach, plum, mango} ????={6 apples, 5 oranges, 10 bananas, 3 pears, 7 peaches, 11 plums, 2 mangos}

Answers

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.

It is known that for all tests administered last​ year, the distribution of scores was approximately normal with mean 74 and standard deviation 7.1. a. A particular employer requires job candidates to score at least 80 on the dexterity test. Approximately what percentage of the test scores during the past year exceeded 80​?

Answers

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%

​Thirty-five math​ majors, 23 music majors and 65 history majors are randomly selected from 586 math​ majors, 224 music majors and 254 history majors at the state university. What sampling technique is​ used?

Answers

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.

Consider the equation of the line, c(c2 + 2)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. A(c) =

Answers

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

Use a Venn diagram to answer the question. A survey of 180 families showed that 67 had a​ dog; 52 had a​ cat; 22 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. How many had a parakeet​ only?

Answers

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

32°


45°


48°


20°

FIND WZC

Answers

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°

r 2 • r • r 5

Multiply or divide as indicated.

Answers

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]

VWX and NOP are similar. If mV = 44° and mP = 66°, what is
mW?

Answers

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°

A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=6−x2. What are the dimensions of such a rectangle with the greatest possible area?

Answers

Answer:

length: 2sqrt(2)

width: 4

Step-by-step explanation:

Let the x-coordinate of the right lower vertex be x. Then the x-coordinate of the left lower vertex is -x. The distance between x and -x is 2x, so the lower side of the rectangle, that rests on the x-axis, measures 2x. The length of the rectangle is 2x.

We now use the equation of the parabola to find the y-coordinate of the upper vertices. y = 6 - 2^x. If you plug in x for x, naturally you get y = 6 - x^2, so the y-coordinates of the two upper vertices are 6 - x^2. Since the y-coordinates of the lower vertices is 0, the vertical sides of the rectangle have length 6 - x^2. The width of the rectangle is 6 - x^2.

We have a rectangle with length 2x and width 6 - x^2.

Now we can find the area of the rectangle.

area = length * width

A = 2x(6 - x^2)

A = 12x - 2x^3

To find a maximum value of x, we differentiate the expression for the area with respect to x, set the derivative equal to zero, and solve for x.

dA/dx = 12 - 6x^2

We set the derivative equal to zero and solve for x.

12 - 6x^2 = 0

2 - x^2 = 0

x^2 = 2

x = +/- sqrt(2)

length = 2x = 2 * sqrt(2) = 2sqrt(2)

width = 6 - x^2 = 6 - (sqrt(2))^2 = 6 - 2 = 4

Final answer:

The greatest possible area of a rectangle inscribed under the parabola y = 6 - x² is achieved with dimensions 2√3 (width) by 3 (height), obtained by optimizing the area function using calculus.

Explanation:

To find the dimensions of the rectangle with the greatest possible area that is inscribed with its base on the x-axis and its upper corners on the parabola y = 6 - x², we apply the optimization technique in calculus.

Let the x-coordinate of the rectangle's upper corners be x, which means that the width of the rectangle is 2x (since the parabola is symmetric with respect to the y-axis). The height of the rectangle is given by the y-coordinate on the parabola, which is 6 - x².

Therefore, the area A of the rectangle can be expressed as A = 2x(6 - x²). To find the maximum area, we take the derivative of A with respect to x and set it to zero, solving for x.

A'(x) = 2(6 - x²) - 4x²(x). Setting A'(x) = 0 gives us x = √3. The maximum area occurs when x = √3, so the width of the rectangle will be 2√3 and the height will be 6 - 3, which simplifies to 3.

Hence, the rectangle with the greatest possible area has dimensions of 2√3 by 3.

The features of a Coverdell Education Savings Account include all of the following EXCEPT: (A) The contributions are deductible. (B) $2,000 is the maximum contribution in any one year. (C) Withdrawals are tax free. (D) Contributions are phased out for certain taxpayers who have adjusted gross income above a certain level.

Answers

Answer:

c

Step-by-step explanation:

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