Those who want a simple tax want no tax, therefore option D is correct answer.
What is tax?Taxes are mandatory payments that citizens and businesses must make to the government, whether they are local, regional, or federal. Tax revenues are used to pay for government programmes like Social Security and Medicare as well as public works and services like roads and schools.
Taxes are paid by whoever is responsible for covering their cost, whether that is the taxed entity, such as a business, or the consumers of the goods that the business produces.
From an accounting perspective, all taxes, including payroll taxes, federal and state income taxes, and sales taxes, must be considered.
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Four students are simplifying the expression 5x(–3 – 7x).
Which student’s work is correct?
5x(–3 – 7x)
(5x)(–3) – 7x
–15x –7x
5x(–3 – 7x)
(5x)(–3) – (5x)(7x)
–15 – 35x
5x(–3 – 7x)
–3 + 5x – 7x
–3 – 2x
5x(–3 – 7x)
(5x)(–3) + (5x)(–7x)
–15x – 35x2
Answer:
So the questions isn't clear enough, but I could tell u that the nearest answer is the last answer, but without the multiplied 2
It's _15x_35xsquared
Answer:
5x(–3 – 7x)
(5x)(–3) + (5x)(–7x)
–15x – 35x2
Step-by-step explanation:
D
claire and martin are trying to determine the length of a slug. Claire has a mesuring stick whose markings are 1 cm apart, and Martin has a measuring stick whose markings are 0.1cm apart.
Claire's measuring stick would make a more precise measurement
How to determine the precise measurement?The given parameters are:
Claire = 1 cm
Martin = 0.1 cm
The 0.1 cm marking would make a more accurate measurement.
This is so because the length of a slug would most likely be in decimal centimeters
However, this doesn't translate to a precise measurement.
Hence, Claire's measuring stick would make a more precise measurement
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find the slope of this line
Answer:
2
Step-by-step explanation:
Select two points that the red line goes through like (0,-10) and (2,-6). If I start at the bottom left of the red line, I can see that I am moving to the right and up. That means that the slope will be positive. As my x is increasing so is my y. Starting at point (0,-10) 2 space, but each space counts as 2, so I really move up 4. Then I check to see how much I moved right. It is only one space, but again, I am counting by 2s, so I really moved 2 units to the right.
The fraction would be 4/2 which just equals 2/1 or 2.
Solve [tex]2 cosx=4cosx sin^2x[/tex]
There are four solutions for the trigonometric equation 2 · cos x = 4 · cos x · sin² x are x₁ = π/4 ± 2π · i, x₂ = 3π/4 ± 2π · i, x₃ = 5π/4 ± 2π · i and x₄ = 7π/4 ± 2π · i, [tex]i \in \mathbb{N}_{O}[/tex].
How to solve a trigonometric equation
In this problem we must simplify the trigonometric equation by both algebraic and trigonometric means and clear the variable x:
2 · cos x = 4 · cos x · sin² x
2 · sin² x = 1
sin² x = 1/2
sin x = ± √2 /2
There are several solutions:
x₁ = π/4 ± 2π · i, [tex]i \in \mathbb{N}_{O}[/tex]
x₂ = 3π/4 ± 2π · i, [tex]i \in \mathbb{N}_{O}[/tex]
x₃ = 5π/4 ± 2π · i, [tex]i \in \mathbb{N}_{O}[/tex]
x₄ = 7π/4 ± 2π · i, [tex]i \in \mathbb{N}_{O}[/tex]
There are four solutions for the trigonometric equation 2 · cos x = 4 · cos x · sin² x are x₁ = π/4 ± 2π · i, x₂ = 3π/4 ± 2π · i, x₃ = 5π/4 ± 2π · i and x₄ = 7π/4 ± 2π · i, [tex]i \in \mathbb{N}_{O}[/tex].
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paulette spent 45.50 on concert tickets before fees. there was a service fee of 6% added on to her purchase.
The original price of concert tickets before the addition of service fee is 42.93
PercentageAmount spent on tickets including service fee = 45.50Percentage if service fee = 6%Original cost before service fee = x45.50 = x + (6% of x)
45.50 = x + (0.06 × x)
45.50 = x + 0.06x
45.50 = 1.06x
x = 45.50/1.06
x = 42.9245283018867
Approximately,
x = 42.93
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6) Ron bought a car 8 years ago for $15,694. If the rate of depreciation is 4.75% per year, what is the value of
Ron's car now? Round answer to nearest cent.
O a.) $10,642.94
Ob.) $10,332.94
Oc.) $10,632.94
Od.) $10,662.94
To the nearest percent how much greater is 8 than 6
The percent greater of 8 than 6 to the nearest percent is 25%
PercentagePercent greater = difference / higher chance value × 100
= 2/8 × 100
= 0.25 × 100
Percent greater = 25%
Therefore, the percent greater of 8 than 6 to the nearest percent is 25%
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Which expression is equivalent to
xay
8
700
8√√√x
y
8√√y
Mark this and return
128x56
√ 2x75
? Assume x > 0 and y> 0.
Save and Exit
The equivalent expression of [tex]\sqrt{\frac{128x^5y^6}{2x^7y^5}}[/tex] is [tex]\frac{8\sqrt y}{x}[/tex]
How to determine the equivalent expression?The expression is given as:
[tex]\sqrt{\frac{128x^5y^6}{2x^7y^5}}[/tex]
Divide 128 by 2
[tex]\sqrt{\frac{64x^5y^6}{x^7y^5}}[/tex]
Apply the law of indices to the variables
[tex]\sqrt{\frac{64y^{6-5}}{x^{7-5}}}[/tex]
Evaluate the differences
[tex]\sqrt{\frac{64y}{x^2}}[/tex]
Take the square root of 64
[tex]8\sqrt{\frac{y}{x^2}}[/tex]
Take the square root of x^2
[tex]\frac{8\sqrt y}{x}[/tex]
Hence, the equivalent expression of [tex]\sqrt{\frac{128x^5y^6}{2x^7y^5}}[/tex] is [tex]\frac{8\sqrt y}{x}[/tex]
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Find the coordinates of the vertices of the triangle if it’s rotated 270° counterclockwise.
If these coordinates are rotated 270° counterclockwise, the resulting coordinates will be;
X' = (3, -1)
Y' = (2, -2)
Z' = (-1, -3)
Rotation of coordinatesIf the coordinates (x, y) is rotated 270° counterclockwise, the resulting coordinate point will be;
(x, y) - > (y, -x)
From the given triangles, the coordinate points are expressed as
X = (1, 3)
Y = (2, 2)
Z = (3, -1)
If these coordinates are rotated 270° counterclockwise, the resulting coordinates will be;
X' = (3, -1)
Y' = (2, -2)
Z' = (-1, -3)
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The total area of a picture and its frame can be represented by ^l + 2f h^w + 2f h, where l and w represent the length and width of the picture, respectively, and f represents the thickness of the frame.
a. Sketch a diagram to represent this scenario.
b. What binomial multiplication represents the area of an 8" # 12" picture and its frame?
c. Multiply the binomials from b., and combine like terms to simplify.
d. What is the total area of an 8" # 12" picture and its frame if the frame is 1.5" thick?
The answers to the question are
b. (l + 2f) (w + 2f)
c. f + 4, f + 6
d. 52.25
a. The diagram is an attachment.
B. How to solve for the binomial multiplicationThe equation says
(l + 2f) (w + 2f)
We have the lengths of w and l as 8 and 12 respectively. Hence we have
(8 + 2f) (12 + 2f) as the area
c. (8 + 2f) (12 + 2f)
= 96 + 16f + 24f + 4f²
= 4f² + 40 f + 96
We have to factorize this using the quadratic equation calculator
f + 4, f + 6
= f = -4 and f = -6
d. If it is 1.25 thick we would have
1.25 x 2 = 2.5
12 - 2.5 = 9.5
8 - 2.5 = 5.5
Area = l * w
= 5.5 * 9.5
= 52.25
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look at the pictures
The key feature that the function of f(x) and g(x) has in common is the domain.
What is the domain and range of a function?The domain of a function is the set of input or argument values for which the function is valid and well defined. The range is the set of the dependent variable for which a function is defined.
From the given information, we are to find the domain, range, x-intercept, and, y-intercept of the given equation:
[tex]\mathbf{f(x) = -4^x+5}[/tex]
[tex]\mathbf{g(x) = x^3 +x^2 -4x+5}[/tex]
For [tex]\mathbf{f(x) = -4^x+5}[/tex];
The domain of a function [tex]\mathbf{-4^x+5}[/tex] has no undefined points or constraints. Thus, the domain is -∞ < x < ∞.
The range f(x) < 5. The x-intercepts, when y is zero = [tex]\mathbf{(\dfrac{In(5)}{2In(2)},0)}[/tex]The y-intercepts; when x is zero = (0,4)For [tex]\mathbf{g(x) = x^3 +x^2 -4x+5}[/tex]
The domain of a function [tex]\mathbf{g(x) = x^3 +x^2 -4x+5}[/tex] has no undefined points or constraints. Thus, the domain is -∞ < x < ∞.
The range -∞ < f(x) < ∞. The x-intercepts, when y is zero = (-2.939, 0)The y-intercepts; when x is zero = (0,5)Learn more about the domain and range of a function here;
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James has $20 in his wallet and makes $10 per hour. Kelly has $50 in her wallet but only makes $8 per hour.
How many hours of work will it take for these two people to have the same amount of money?
Answer:
At 15 hours of work they will be equal.
Step-by-step explanation:
James: y = 10x + 20
Kelly: y = 8X + 50
Set them equal to each other and solve
10x + 20 = 8x + 50 Subtract 8x from both sides of the equation
2x + 20 = 50 Subtract 20 from both sides of the equation
2x = 30 Divide both sides by 2
x =15
inverse function of f(x)=x-7/x+4
Final Answer: The inverse of f (x)=7x-4 is f^-1 (x)= (x+4)/7
where is the center of the circle and the radius is how many units?
Answer:
8,9,5
Step-by-step explanation:
a 300-acre farm produced 20,000 bushels of corn last year. what is the minimum rate of production, in bushels per acre, that is needed this year, so the farm's two-year production total will be at least 49,100 bushels of corn?
The minimum production rate of bushels per acre of corn needed this year is 97 for two year's production to reach the goal of 49,100.
How to calculate the minimum production rate?To calculate the minimum production rate for this year, we must subtract the production rate of the previous year, with what is expected to be obtained this year.
49,100 - 20,000 = 29,100Then we must divide the minimum that must be obtained by the total area, by the 300 acres that we have, to know how much corresponds to each acre.
29,100 ÷ 300 = 97According to the above, each acre must produce 97 bushels of corn to reach the goal of 49,100 for both years.
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Find the exact solutions of x2 − 3x − 5 = 0 using the quadratic formula. Show all work! 75 points please help!!!!!
Answer:
[tex]x=\dfrac{3+ \sqrt{29}}{2}, \quad \dfrac{3- \sqrt{29}}{2}[/tex]
Step-by-step explanation:
Quadratic Formula
[tex]x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\quad\textsf{when }\:ax^2+bx+c=0[/tex]
Given quadratic equation:
[tex]x^2-3x-5=0[/tex]
Define the variables:
[tex]\implies a=1, \quad b=-3, \quad c=-5[/tex]
Substitute the defined variables into the quadratic formula and solve for x:
[tex]\implies x=\dfrac{-(-3) \pm \sqrt{(-3)^2-4(1)(-5)}}{2(1)}[/tex]
[tex]\implies x=\dfrac{3 \pm \sqrt{9+20}}{2}[/tex]
[tex]\implies x=\dfrac{3 \pm \sqrt{29}}{2}[/tex]
Therefore, the exact solutions to the given quadratic equation are:
[tex]x=\dfrac{3+ \sqrt{29}}{2}, \quad \dfrac{3- \sqrt{29}}{2}[/tex]
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Answer:
[tex]\boxed {\frac{3+\sqrt{29}}{2}} \boxed{\frac{3-\sqrt{29}}{2}}[/tex]
Step-by-step explanation:
Quadratic Formula :
[tex]\boxed {\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}}[/tex]
We are given the equation x² - 3x - 5 = 0.
Here,
a = 1b = -3c = -5Solving :
3 ± √3² - 4(1)(-5) / 2(1)3 ± √29 / 2Hence, the solutions are :
[tex]\boxed {\frac{3+\sqrt{29}}{2}} \boxed{\frac{3-\sqrt{29}}{2}}[/tex]
There are some 1-dollar bills, some 5-dollar bills, and a 20-dollar bill in an envelope. The number of 1-dollar bills is five times the number of the 5-dollar bills. We randomly pull a bill from the envelope. How many 1-dollar bills are in this envelope if the expected value of the bill pulled is $2.00?
Answer:
the number of 1 dollars bills was 5 si make 1 more 5 dollars so 8n total it is 10 dollars.
A solid right square prism is cut into 5 equal pieces parallel to its bases. The volume of each piece is V = 1/5s2h.Solve the formula for s. help asap
The solution of the formula for s is:
[tex]s = \sqrt{\frac{5V}{h}}[/tex]
What is the symbolic equation?The equation is:
[tex]V = \frac{1}{5}s^2h[/tex]
To solve for s, we isolate it, hence:
[tex]s^2 = \frac{5V}{h}[/tex]
[tex]s = \sqrt{\frac{5V}{h}}[/tex]
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What point is 2/3 of the distance from point A(3, 1) to point B(3, 19)?
What point is 2/3 of the distance from point A(3, 1) to point B(3, 19)?
(3, 13)
(9/3, 13)
(2, 12)
(3, 19)
Answer: (3, 13)
Step-by-step explanation:
If we let the point be P, then AP:BP=2:1.
[tex]P=\left(\frac{(2)(3)+(1)(3)}{2+1}, \frac{(2)(19)+(1)(1)}{2+1} \right)=(3, 13)[/tex]
Using the drawing what is the vertex of angle 4?
Answer: A
Step-by-step explanation:
Suppose you have $1500 in your savings account at the end of a certain period of time. You invested $1000 at a 2.71% simple annual interest rate. How long, in years, was your money invested? State your result to the nearest hundredth of a year.
The time taken in years for money invested to accumulate to $1500 is 18.45 years
Simple interestPrincipal = $1000Interest rate = 2.71% = 0.0271t = ?Simple interest = Total savings - principal
= 1500 - 1000
= $500
Simple interest = principal × rate × time
500 = 1000 × 0.0271 × t
500 = 27.1 × t
500 = 27.1t
t = 18.4501845018450
Approximately,
t = 18.45 years
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Hi I would like to know how I can solve this problem.
The [tex]n[/tex]-th term is
[tex]U_n = \dfrac14 n^2 (n+1)^2[/tex]
so the 39th term is
[tex]U_{39} = \dfrac14 39^2 40^2 = \boxed{608,400}[/tex]
Observe that
[tex]2^3 + 4^3 + 6^3 = 2^3 + 2^3\times2^3 + 2^3\times3^3 = 8\left(1^3+2^3+3^3)[/tex]
which suggests that
[tex]V_n = 8U_n = \boxed{2n^2(n+1)^2}[/tex]
Complete the equations to solve 1{,}860 \div61,860÷61, comma, 860, divided by, 6.
\phantom{=}\greenD{1{,}860}\div{\blueD6}=1,860÷6empty space, start color #1fab54, 1, comma, 860, end color #1fab54, divided by, start color #11accd, 6, end color #11accd
=(\greenD{1{,}800}\div\,=(1,800÷equals, left parenthesis, start color #1fab54, 1, comma, 800, end color #1fab54, divided by
) \, + \,(\greenD{60}\div\,)+(60÷right parenthesis, plus, left parenthesis, start color #1fab54, 60, end color #1fab54, divided by
))right parenthesis
= 300 +=300+equals, 300, plus
==equals
The division of the figure based on the information is 14.09.
How to illustrate the information?It should be noted that the question is simply to divide 860 by 61.
The division based on the information will be illustrated thus:
= 860 ÷ 61
= 14.09
In conclusion, the correct option is 14.09.
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Suppose f(x) and g(x) are differentiable functions satisfying f(1) = f′(1) = 2. Let
h(x) = g(f(x)) − g(2x). Determine whether h′(1) > 1
Answer:
1777777777888876332222233
Please look at the attachment and answer my question :(
Answer:
AB / BC = 2 / 3
A equals 9 and C = 13
AC = 13 - 9 = 4
AB + BC = 13
A) AB + BC = 4
B) AB / BC = 2/3 Therefore BC = AB / (2/3)
B) BC = 1.5 AB
A) BC = 4 - A/B
Multiplying equation A) by -1
A) -BC = -4 + AB then we add equation B)
B) BC = 1.5 AB then adding both equations
0 = 2.5 AB -4
2.5 AB = 4
AB = 1.6
Since A = 9 then the number at B is 9 + 1.6
equals 10.6
Step-by-step explanation:
Answer:
The number at B is 10.6
Step-by-step explanation:
Let the number at B be x.
We know that distance between two points on the number line is the absolute value of the difference of numbers at those points.
Then distances representing the lengths of segments AB and BC are:
AB = x - 9BC = 13 - xWe are given the ratio of segments:
AB/BC = 2/3Substitute and solve for x:
(x - 9)/(13 - x) = 2/33(x - 9) = 2(13 - x)3x - 27 = 26 - 2x3x + 2x = 26 + 275x = 53x = 53/5x = 10.6Find the sum of the geometric series given a1 = 5, r = 3, n = 12.
The sum of the geometric series is 1328600
How to determine the sum?The given parameters are:
a1 = 5
r = 3
n = 12
The sum of the terms is:
[tex]S_n = \frac{a(1 - r^n)}{1 - r}[/tex]
This gives
[tex]S_n = \frac{5 * (1 - 3^{12})}{1 - 3}[/tex]
Evaluate the difference
[tex]S_n = \frac{5 * (1 - 531441)}{-2}[/tex]
This gives
[tex]S_n = \frac{5 * - 531440}{-2}[/tex]
Evaluate
[tex]S_n = 1328600[/tex]
Hence, the sum of the geometric series is 1328600
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in triangle ABC, AC=13, BC=84, and AB=85. find the measure of angle C.
in triangle ABC, AC=13, BC=84, and AB=85. the measure of angle is C.5.1 degrees,
How can the angle be found?Using the The Law of Cosines in any triangle which can be expressed as;
[tex]c^2 = a^2 + b^2 - 2ab * cos(C)[/tex]
Then if we expressed the given sides in the cosine formula we have
[tex]13^2 = 85^2 + 84^2 - 2 * 85 * 84 * cos(C)[/tex]
Then we have [tex]169 = 7225 + 7056 - 14280 * cos(C)[/tex]
[tex]14280 * cosC = (7225 + 7056 - 169)[/tex]
[tex]cosC = \frac{14212}{ 14280}[/tex]
cos 0.9947
[tex]C = cos^-1(0.9947)[/tex]
C = 5.1 degrees
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What is the solution to the following system of equations?
x-3y=6
2x + 2y = 4
(-1,3)
(3,-1)
(1, -3)
(-3,1)
Answer:
(b) (3,-1)
Step-by-step explanation:
One can determine the correct answer by checking to see if it satisfies the equations.
CheckingThe first equation can be rewritten to ...
x = 6 +3y . . . . . . . add 3y to both sides
This makes it easier to check answer choices:
a) -1 = 6 +3(3) . . . . no
b) 3 = 6 +3(-1) . . . . yes . . . . . (3, -1) is the solution
c) 1 = 6 +3(-3) . . . . no
d) -3 = 6 +3(1) . . . . no
__
Additional comment
We can further convince ourselves this is the correct choice by seeing if it satisfies the second equation:
2x +2y = 4 . . . for (x, y) = (3, -1)
2(3) +2(-1) = 4 . . . . yes
suppose that fanf g are continuous functions,
12_(^28 f(x)dx=19, and 12 (^28 g(x)dx =35.
Find
12_ f^28 [kg(x)-2f(x)] dx, where the coefficient
k=7
those are intergral symbols with numbers on
top and bottom. please show work. thanks
The required result based on given continuous functions is: 207. See the explanation for same below.
A continuous function in mathematics is one in which a continuous variation (that is, a change without a jump) of the argument causes a continuous variation of the function's value.
Since G and F are continuous functions,
[tex]\int_{12}^{28}) \, f(x) dx[/tex] = 19
[tex]\int_{12}^{28}) \, f(x) dx[/tex] = 35
Therefore,
[tex]\int_{12}^{28}) \, [K g(x) - 2 f(x)] dx[/tex]
= K [tex]\int_{12}^{28}) \, g(x) dx -2 \int_{12}^{28}) \, f(x) dx[/tex]
So, given that K = 7, we have
7 x 35 - 2 x 19
= 245 - 38
= 207
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A random variable X has a gamma density function with parameters α= 8 and β = 2.
Without making any assumptions, derive the moment generating function of X and use to
determine the mean and variance of X.
I know you said "without making any assumptions," but this one is pretty important. Assuming you mean [tex]\alpha,\beta[/tex] are shape/rate parameters (as opposed to shape/scale), the PDF of [tex]X[/tex] is
[tex]f_X(x) = \dfrac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x} = \dfrac{2^8}{\Gamma(8)} x^7 e^{-2x}[/tex]
if [tex]x>0[/tex], and 0 otherwise.
The MGF of [tex]X[/tex] is given by
[tex]\displaystyle M_X(t) = \Bbb E\left[e^{tX}\right] = \int_{-\infty}^\infty e^{tx} f_X(x) \, dx = \frac{2^8}{\Gamma(8)} \int_0^\infty x^7 e^{(t-2) x} \, dx[/tex]
Note that the integral converges only when [tex]t<2[/tex].
Define
[tex]I_n = \displaystyle \int_0^\infty x^n e^{(t-2)x} \, dx[/tex]
Integrate by parts, with
[tex]u = x^n \implies du = nx^{n-1} \, dx[/tex]
[tex]dv = e^{(t-2)x} \, dx \implies v = \dfrac1{t-2} e^{(t-2)x}[/tex]
so that
[tex]\displaystyle I_n = uv\bigg|_{x=0}^{x\to\infty} - \int_0^\infty v\,du = -\frac n{t-2} \int_0^\infty x^{n-1} e^{(t-2)x} \, dx = -\frac n{t-2} I_{n-1}[/tex]
Note that
[tex]I_0 = \displaystyle \int_0^\infty e^{(t-2)}x \, dx = \frac1{t-2} e^{(t-2)x} \bigg|_{x=0}^{x\to\infty} = -\frac1{t-2}[/tex]
By substitution, we have
[tex]I_n = -\dfrac n{t-2} I_{n-1} = (-1)^2 \dfrac{n(n-1)}{(t-2)^2} I_{n-2} = (-1)^3 \dfrac{n(n-1)(n-2)}{(t-2)^3} I_{n-3}[/tex]
and so on, down to
[tex]I_n = (-1)^n \dfrac{n!}{(t-2)^n} I_0 = (-1)^{n+1} \dfrac{n!}{(t-2)^{n+1}}[/tex]
The integral of interest then evaluates to
[tex]\displaystyle I_7 = \int_0^\infty x^7 e^{(t-2) x} \, dx = (-1)^8 \frac{7!}{(t-2)^8} = \dfrac{\Gamma(8)}{(t-2)^8}[/tex]
so the MGF is
[tex]\displaystyle M_X(t) = \frac{2^8}{\Gamma(8)} I_7 = \dfrac{2^8}{(t-2)^8} = \left(\dfrac2{t-2}\right)^8 = \boxed{\dfrac1{\left(1-\frac t2\right)^8}}[/tex]
The first moment/expectation is given by the first derivative of [tex]M_X(t)[/tex] at [tex]t=0[/tex].
[tex]\Bbb E[X] = M_x'(0) = \dfrac{8\times\frac12}{\left(1-\frac t2\right)^9}\bigg|_{t=0} = \boxed{4}[/tex]
Variance is defined by
[tex]\Bbb V[X] = \Bbb E\left[(X - \Bbb E[X])^2\right] = \Bbb E[X^2] - \Bbb E[X]^2[/tex]
The second moment is given by the second derivative of the MGF at [tex]t=0[/tex].
[tex]\Bbb E[X^2] = M_x''(0) = \dfrac{8\times9\times\frac1{2^2}}{\left(1-\frac t2\right)^{10}} = 18[/tex]
Then the variance is
[tex]\Bbb V[X] = 18 - 4^2 = \boxed{2}[/tex]
Note that the power series expansion of the MGF is rather easy to find. Its Maclaurin series is
[tex]M_X(t) = \displaystyle \sum_{k=0}^\infty \dfrac{M_X^{(k)}(0)}{k!} t^k[/tex]
where [tex]M_X^{(k)}(0)[/tex] is the [tex]k[/tex]-derivative of the MGF evaluated at [tex]t=0[/tex]. This is also the [tex]k[/tex]-th moment of [tex]X[/tex].
Recall that for [tex]|t|<1[/tex],
[tex]\displaystyle \frac1{1-t} = \sum_{k=0}^\infty t^k[/tex]
By differentiating both sides 7 times, we get
[tex]\displaystyle \frac{7!}{(1-t)^8} = \sum_{k=0}^\infty (k+1)(k+2)\cdots(k+7) t^k \implies \displaystyle \frac1{\left(1-\frac t2\right)^8} = \sum_{k=0}^\infty \frac{(k+7)!}{k!\,7!\,2^k} t^k[/tex]
Then the [tex]k[/tex]-th moment of [tex]X[/tex] is
[tex]M_X^{(k)}(0) = \dfrac{(k+7)!}{7!\,2^k}[/tex]
and we obtain the same results as before,
[tex]\Bbb E[X] = \dfrac{(k+7)!}{7!\,2^k}\bigg|_{k=1} = 4[/tex]
[tex]\Bbb E[X^2] = \dfrac{(k+7)!}{7!\,2^k}\bigg|_{k=2} = 18[/tex]
and the same variance follows.