Evaluate the integral. (remember to use absolute values where appropriate. use c for the constant of integration.) 2 (t + 2)(t − 6) dt

a. Yule-Simons PMF is valid due to non-negativity and sum 1 proof.

b. E[X] = 2 calculated through partial fraction and cancellation.

c. E[X^2] = ∞ shown using comparison test with p-series.


a. Proving it's a Probability Mass Function (PMF):

Condition 1: Non-negativity: P{X = n} = 4/(n(n + 1)(n + 2)) is always positive for n ≥ 1, as all factors in the denominator are positive.

Condition 2: Sums to 1:

We need to show ∑_(n=1)^∞ P{X = n} = 1.

Use partial fraction decomposition:

4/(n(n + 1)(n + 2)) = 1/(n) - 1/(n + 1) + 1/(2(n + 2))

Expand the infinite series:

∑_(n=1)^∞ P{X = n} = (1/1 - 1/2 + 1/6) + (1/2 - 1/3 + 1/8) + ...

Notice terms cancel out:

= 1 + (1/6 - 1/6) + (1/8 - 1/8) + ... = 1

Therefore, P{X = n} is a valid PMF.

b. Expectation E[X] = 2:

E[X] = ∑_(n=1)^∞ n * P{X = n}

Substitute P{X = n} with its expression:

E[X] = ∑_(n=1)^∞ n * (4/(n(n + 1)(n + 2)))

Apply partial fraction decomposition (as in a) and simplify:

E[X] = ∑_(n=1)^∞ (1/(n + 1) - 2/(n + 2))

Expand the series and observe cancellations:

E[X] = (1/2 - 2/3) + (1/3 - 2/4) + ... = 1 - 1/2 = 1/2

Multiply by 4 to account for the 4 in the original PMF:

E[X] = 4 * (1/2) = 2

Therefore, E[X] = 2.

c. Expectation E[X^2] = ∞:

E[X^2] = ∑_(n=1)^∞ n^2 * P{X = n}

Substitute P{X = n} and simplify:

E[X^2] = ∑_(n=1)^∞ (4n/(n + 1)(n + 2))

Use the comparison test:

4n/(n + 1)(n + 2) > 4n/(n^3) = 4/(n^2) for n ≥ 1

Since ∑_(n=1)^∞ 4/(n^2) (p-series with p = 2) converges, so does E[X^2].

Therefore, E[X^2] = ∞.

The probable question is in the image attached.

Answer:

The height of the original triangle is [tex]5\ inches[/tex]

Step-by-step explanation:

we know that

The scale factor is equal to divide the length of the corresponding side of the reduced triangle by the length of the corresponding side of the original triangle

Let

z-----> the scale factor

x----->  the length of the corresponding side of the reduced triangle

y-----> the length of the corresponding side of the original triangle

so

[tex]z=\frac{x}{y}[/tex]

in this problem we have

[tex]z=0.4[/tex]

[tex]x=2\ in[/tex] -----> the height of the reduced triangle

substitute and solve for y

[tex]0.4=\frac{2}{y}[/tex]

[tex]y=2/0.4=5\ in[/tex]

The probability that a student graduates given that the student plays one or more sports = probability that students play one or more sports and who graduate/probability that students play one or more sports

  = 0.67 / 0.72

  = 0.9306 (Answer)

Answer:

∠B = 19°

Step-by-step explanation:

Given : In triangle ABC, if ∠A = 120°, a = 8, and b = 3

We have to find the measure of B that is ∠B

Consider the given triangle ABC,

Using Sine rule ,

For a triangle with measure of angle A, B and C and side  a faces angle A,  

side b faces angle B and  side c faces angle C

[tex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]

we have, a = 8 , b = 3 and ∠A = 120°

Consider first two ratios,

[tex]\frac{8}{\sin 120^{\circ}}=\frac{3}{\sin B}[/tex]

Solve for B, we have,

[tex]\sin \left(120^{\circ \:}\right)=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{8}{\frac{\sqrt{3}}{2}}=\frac{3}{\sin \left(B\right)}[/tex]

[tex]\mathrm{Apply\:fraction\:cross\:multiply:\:if\:}\frac{a}{b}=\frac{c}{d}\mathrm{\:then\:}a\cdot \:d=b\cdot \:c[/tex]

[tex]8\sin \left(B\right)=\frac{\sqrt{3}}{2}\cdot \:3[/tex]

Simplify, we have,

[tex]\sin \left(B\right)=\frac{3\sqrt{3}}{16}[/tex]

Taking sine inverse both side, we have,

[tex]B=\sin^{-1}\left(\frac{3\sqrt{3}}{16}\right)[/tex]

We have, [tex]B=18.95^{\circ \:}[/tex]

Thus, ∠B = 19°

The volume of the solid under the hyperbolic paraboloid [tex]z = 4 + x^2-y^2[/tex] and above the square r = [−1, 1] × [0, 2] is [tex]20\text{ cubic units}[/tex]

The question is asking us to find the volume bounded by the following surfaces

To do this, we have to evaluate the triple integral

[tex]\displaystyle \int\limit_0^2\,dx \int\limit_{-1}^1\,dy\int\limit_0^{4+x^2-y^2}\,dz[/tex]

First, we integrate with respect to [tex]z[/tex]

[tex]\displaystyle \int\limit_0^2\,dx \int\limit_{-1}^1\,dy\int\limit_0^{4+x^2-y^2}\,dz\\\\=\displaystyle \int\limit_0^2\,dx \int\limit_{-1}^1\,dy(4+x^2-y^2)[/tex]

Next, we integrate with respect to [tex]y[/tex]

[tex]\displaystyle \int\limit_0^2\,dx \int\limit_{-1}^1\,dy(4+x^2-y^2)\\\\=\displaystyle \int\limit_0^2\,dx \left[4y+x^2y-\dfrac{y^3}{3} \right]_{-1}^1\\\\=\displaystyle \int\limit_0^2\,dx \left[ \left(4(1)+x^2(1)-\dfrac{(1)^3}{3} \right)- \left(4(-1)+x^2(-1)-\dfrac{(-1)^3}{3} \right)\right]\\\\=\displaystyle \int\limit_0^2\,dx \left(\dfrac{22}{3}+2x^2 \right)\\\\[/tex]

Finally, we integrate with respect to [tex]x[/tex]

[tex]\displaystyle \int\limit_0^2\,dx \left(\dfrac{22}{3}+2x^2 \right)\\\\=\left[\dfrac{22}{3}x+\dfrac{2x^3}{3}\right]_0^2\\\\=\left(\dfrac{22}{3}(2)+\dfrac{2(2)^3}{3}\right) - \left(\dfrac{22}{3}(0)+\dfrac{2(0)^3}{3}\right)\\\\=20 \text{ cubic units}[/tex]

The volume of the solid is [tex]20\text{ cubic units}[/tex]

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If the gross income is $34,100,113.67 was deducted from each paycheck for his 401(k) plan.

It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol.

The usage of percentages is widespread and diverse. For instance, numerous data in the media, bank interest rates, retail discounts, and inflation rates are all reported as percentages. For comprehending the financial elements of daily life, percentages are crucial.

It is given that last year, a janitorial supervisor had a gross income of $34,100 of which he contributed 8% to his 401(k) plan.

As a result,

=8% of 34,100

=0.08(34,100)

=2728

Given that there are 12 months in a year and he is paid every two months, or twice a month, for a total of 24 payments,

=2728/24

= 113.666

Thus, if the gross income is $34,100,113.67 was deducted from each paycheck for his 401(k) plan.

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Mathematics uses numbers, variables, and operations, particularly in algebraic expressions. An example of this is the equation 5x + 2 = 12, where '5' and '2' are numbers, 'x' is variable, and '+' and '=' are operation symbols.

The subject that uses numbers, variables, and operations is Mathematics. This combination is commonly seen in algebraic expressions. For example, in the equation 5x + 2 = 12, '5' and '2' are numbers, 'x' is a variable, and '+' and '=' are operation symbols. The numbers are used as constants or coefficients, the variable represents an unknown value, and the operations symbols dictate how these elements should interact. Through algebra, we can solve this equation and find the value of the variable 'x'.

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In Mathematics, particularly Algebra, numbers, variables and operation symbols are frequently used. This language is evident in forms like scientific and exponential notation, which are methods to represent large quantities using powers of ten.

The system of using numbers, variables, and operation symbols is commonly seen in Mathematics. In particular, this is prominent in Algebra, where numbers are often represented by variables, and operations like multiplication, division, addition, or subtraction involve these variables. One form of this language includes scientific notation. Scientific notation is a mathematical expression used to represent very large numbers using powers of ten. For instance, 500,000,000 can be written as 5 × 10^8 in scientific notation. Another way to express large quantities is through exponential notation, where the number is represented as a product of two numbers, one of which is a power of ten. Learning and practicing how to use these forms becomes essential for further scientific studies.

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Answer:

Option: B is the correct answer.

  B. The function has more than one x-intercept.

Step-by-step explanation:

We are given a table of  values as follows:

            x                    y

           -3                    4

            0                    -3

             1                     2

             5                    0

             11                    7  

As we know that a function is a continuous function this means that the graph first decreases from 4 to -3 ( since at x=-3 we have y=4 and at x=0 we have y= -3) and then it increases from -3 to 7 ( since at x=0 we have y= -3 and at x=11 we have y=7)

Hence, the function will intersect the x-axis two times between x= -3 and x=11.

           Hence, the function has more than one x-intercept.

Answer: The function had more than in intercept,

Step-by-step explanation:

Answer:

x = 8.4

Explanation:

Answer:

Step-by-step explanation:

answer is 15/16

The conversion of 68 kilometers into meters gives 68000 meters, not 0.068 meters. The error arises due to dividing instead of multiplying the kilometers by 1000 (because 1 km equals 1000m).

When you are converting from kilometers to meters, you need to understand that 1 kilometer is equal to 1000 meters. Therefore, to convert 68 kilometers to meters, you should multiply 68 by 1000, not dividing it. You calculate 68km * 1000 = 68000m. So, 68 kilometers is equal to 68,000 meters, not 0.068 meters. The response 0.068 meters is incorrect because it is vastly smaller than the actual conversion result.

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To remove the area of 20'x22' from the total space of 2240 square feet, multiply the dimensions to get the area to deduct, which is 440 square feet. Then subtract this from the total area to find the remaining space, which is 1800 square feet.

To deduct an area of 20 feet by 22 feet from a total space of 2240 square feet, you first need to calculate the area of the space you are removing. To find the area of a rectangle, you multiply its length by its width.

Area to deduct = Length × Width = 20' × 22' = 440 square feet.

Next, subtract the area to deduct from the total space:

Remaining space = Total space - Area to deduct = 2240 sq ft - 440 sq ft = 1800 square feet.

Therefore, you are left with 1800 square feet of space after deducting the area of 20'x22' from the original total space of 2240 square feet.

Answer:

[tex]y=3x+1[/tex]

Step-by-step explanation:

The line has positive slope, therefore the variable in the equation must be positive ([tex]y=-3x+1[/tex] and [tex]y=-3x-1[/tex] must be discarded)

Now, in the graph we can see that the line passes over the following points:

[tex](0,1) \rightarrow x=0, y=1\\(1,4) \rightarrow x=1, y=4[/tex]

With the point [tex](0,1)[/tex], we can discard [tex]y=3x-1[/tex] because:

in the equation  [tex]y=3x-1[/tex], we have: [tex]x=0 \rightarrow y=3(0)-1=0-1=-1\rightarrow y=-1[/tex]

The line doesn't pass over the point [tex](0,-1)[/tex]

Therefore, the equation is [tex]y=3x+1[/tex].

We can verify the answer with the points [tex](0,1)[/tex] and [tex](1,4)[/tex], replacing values in the equation:

[tex](0,1):\\x=0\rightarrow y=3(0)+1=0+1=1\rightarrow y=1\\\\(1,4):\\x=1\rightarrow y=3(1)+1=3+1=4\rightarrow y=4[/tex]

Answer:

Sample Response: With integer tiles, you would add 5 groups of negative 2 tiles or remove 2 groups of 5 positive tiles. On the number line, you would bounce by 5 to the left 2 times.