1. p.d.f. of x_min: f(x_min) = 1/l for 0 ≤ x_min ≤ l. 2. p.d.f. of x_mu: f(x_mu) = δ(x_{mu - μ), where δ represents the delta function. 3. p.d.f. of sample median x: It can be simulated using statistical software or programming.
To find the probability density functions (p.d.f.'s) of x_min, x_mu, and the sample median x, we need to understand their definitions.
1.x_min (minimum value): The p.d.f. of x_min can be found by using the cumulative distribution function (c.d.f.) of the uniform distribution.
The c.d.f. of the uniform distribution on the interval [0, l] is given by F(x) = (x - 0)/(l - 0) = x/l for 0 ≤ x ≤ l. Then, taking the derivative of the c.d.f., we get the p.d.f. of x_min as f(x_min = d F(x_min/dx = 1/l for 0 ≤ x_min ≤ l.
2. x_mu (population mean): Since the population mean (μ) is fixed, the p.d.f. of x_mu is a delta function, which is a spike at the value of μ. Therefore, the p.d.f. of x_mu is
f(x_mu) = δ(x_mu - μ), where δ represents the delta function.
3. Sample median (x): The sample median can be obtained by arranging the observations in ascending order and selecting the middle value. Since we have 9 observations, the sample median will be the 5th value when they are arranged in ascending order.
To find its p.d.f., we need to consider the distribution of the 5th order statistic. Since the sample is from the uniform distribution, the p.d.f. of the 5th order statistic can be found using the formula for the p.d.f. of order statistics.
However, since it involves complicated calculations, it would be easier to simulate the distribution of the sample median using statistical software or programming.
To summarize:
1. p.d.f. of x_min: f(x_min) = 1/l for 0 ≤ x_min ≤ l.
2. p.d.f. of x_mu: f(x_mu) = δ(x_{mu - μ), where δ represents the delta function.
3. p.d.f. of sample median x: It can be simulated using statistical software or programming.
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For what values of x does f(x)=2sinx+sin 2 x have a horizontal tangent?
After finding the values of x, we can substitute them back into the original function f(x) = 2sin(x) + sin(2x) to verify if the tangents are indeed horizontal at those points.
To find the values of x for which the function f(x) = 2sin(x) + sin(2x) has a horizontal tangent, we need to find the critical points of the function where the derivative is equal to zero.
First, let's find the derivative of f(x):
f'(x) = 2cos(x) + 2cos(2x)
To find the critical points, we set the derivative equal to zero and solve for x:
2cos(x) + 2cos(2x) = 0
Now, let's solve this equation. We can start by factoring out 2:
2(cos(x) + cos(2x)) = 0
For the derivative to be zero, either cos(x) + cos(2x) = 0 or the coefficient 2 is zero. Since the coefficient 2 is not zero, we focus on solving cos(x) + cos(2x) = 0.
Using the trigonometric identity cos(2x) = 2cos^2(x) - 1, we can rewrite the equation as:
cos(x) + 2cos^2(x) - 1 = 0
Rearranging the terms, we have:
2cos^2(x) + cos(x) - 1 = 0
Let's solve this quadratic equation for cos(x) using factoring or the quadratic formula. Once we find the values of cos(x), we can determine the corresponding values of x by taking the inverse cosine (arccos) of those values.
After finding the values of x, we can substitute them back into the original function f(x) = 2sin(x) + sin(2x) to verify if the tangents are indeed horizontal at those points.
Please note that solving the quadratic equation may involve complex solutions, and those values of x will not correspond to horizontal tangents.
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Find the real zeros of f. Use the real zeros to factor f. f(x)=x 3
+6x 2
−9x−14 The real zero(s) of f is/are (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) Use the real zero(s) to factor f. f(x)= (Factor completely. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
The real zeros of f are -7, 2, and -1.
To find the real zeros of f(x) = x³ + 6x² - 9x - 14. We can use Rational Root Theorem to solve this problem.
The Rational Root Theorem states that if the polynomial function has any rational zeros, then it will be in the form of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. The constant term of the given function is -14 and the leading coefficient is 1. The possible factors of p are ±1, ±2, ±7, and ±14. The possible factors of q are ±1. The possible rational zeros of the function are: ±1, ±2, ±7, ±14
We can try these values in the given function and see which one satisfies it.
On trying these values we get, f(-7) = 0
Hence, -7 is a zero of the function f(x).
To find the other zeros, we can divide the function f(x) by x + 7 using synthetic division.
-7| 1 6 -9 -14 | 0 |-7 -7 1 -14 | 0 1 -1 -14 | 0
Therefore, x³ + 6x² - 9x - 14 = (x + 7)(x² - x - 2)
We can factor the quadratic expression x² - x - 2 as (x - 2)(x + 1).
Therefore, f(x) = x³ + 6x² - 9x - 14 = (x + 7)(x - 2)(x + 1)
The real zeros of f are -7, 2, and -1 and the factored form of f is f(x) = (x + 7)(x - 2)(x + 1).
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Realize the systems below by canonic direct, series, and parallel forms. b) H(s) = s^3/(s+1)(s²+4s+13)
The transfer function H(s) = s^3/(s+1)(s^2+4s+13) can be realized in the canonic direct, series, and parallel forms.
To realize the given transfer function H(s) = s^3/(s+1)(s^2+4s+13) in the canonic direct, series, and parallel forms, we need to factorize the denominator and express it as a product of first-order and second-order terms.
The denominator (s+1)(s^2+4s+13) is already factored, with a first-order term s+1 and a second-order term s^2+4s+13.
1. Canonic Direct Form:
In the canonic direct form, each term in the factored form is implemented as a separate block. Therefore, we have three blocks for the three terms: s, s+1, and s^2+4s+13. The output of the first block (s) is connected to the input of the second block (s+1), and the output of the second block is connected to the input of the third block (s^2+4s+13). The output of the third block gives the overall output of the system.
2. Series Form:
In the series form, the numerator and denominator are expressed as a series of first-order transfer functions. The numerator s^3 can be decomposed into three first-order terms: s * s * s. The denominator (s+1)(s^2+4s+13) remains as it is. Therefore, we have three cascaded blocks, each representing a first-order transfer function with a pole or zero. The first block has a pole at s = 0, the second block has a pole at s = -1, and the third block has poles at the roots of the quadratic equation s^2+4s+13 = 0.
3. Parallel Form:
In the parallel form, each term in the factored form is implemented as a separate block, similar to the canonic direct form. However, instead of connecting the blocks in series, they are connected in parallel. Therefore, we have three parallel blocks, each representing a separate term: s, s+1, and s^2+4s+13. The outputs of these blocks are summed together to give the overall output of the system.
These are the realizations of the given transfer function H(s) = s^3/(s+1)(s^2+4s+13) in the canonic direct, series, and parallel forms. The choice of which form to use depends on the specific requirements and constraints of the system.
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If 2x+y=9, what is the smallest possible value of 4x 2 +3y 2 ?
The smallest possible value of [tex]4x^2 + 3y^2[/tex] is 64.
To find the smallest value of [tex]4x^2 + 3y^2[/tex]
use the concept of the Arithmetic mean-Geometric mean inequality. AMG inequality states that, for non-negative a, b, have the inequality, (a + b)/2 ≥ √(ab)which can be written as
[tex](a + b)^2/4 \geq ab[/tex]
Equality is achieved if and only if
a/b = 1 or a = b
apply AM-GM inequality on
[tex]4x^2[/tex] and [tex]3y^24x^2 + 3y^2 \geq 2\sqrt {(4x^2 * 3y^2 )}\sqrt{(4x^2 * 3y^2 )} = 2 * 2xy = 4x*y4x^2 + 3y^2 \geq 8xy[/tex]
But xy is not given in the question. Hence, get xy from the given equation
2x + y = 9y = 9 - 2x
Now, substitute the value of y in the above equation
[tex]4x^2 + 3y^2 \geq 4x^2 + 3(9 - 2x)^2[/tex]
Simplify and factor the expression,
[tex]4x^2 + 3y^2 \geq 108 - 36x + 12x^2[/tex]
rewrite the above equation as
[tex]3y^2 - 36x + (4x^2 - 108) \geq 0[/tex]
try to minimize the quadratic expression in the left-hand side of the above inequality the minimum value of a quadratic expression of the form
[tex]ax^2 + bx + c[/tex]
is achieved when
x = -b/2a,
that is at the vertex of the parabola For
[tex]3y^2 - 36x + (4x^2 - 108) = 0[/tex]
⇒ [tex]y = \sqrt{((36x - 4x^2 + 108)/3)}[/tex]
⇒ [tex]y = 2\sqrt{(9 - x + x^2)}[/tex]
Hence, find the vertex of the quadratic expression
[tex](9 - x + x^2)[/tex]
The vertex is located at
x = -1/2, y = 4
Therefore, the smallest value of
[tex]4x^2 + 3y^2[/tex]
is obtained when
x = -1/2 and y = 4, that is
[tex]4x^2 + 3y^2 \geq 4(-1/2)^2 + 3(4)^2[/tex]
= 16 + 48= 64
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7. [Show all stepsl Otherwise, no credit will be awarded.] (20 points) (a) Set up a system of linear equations to determine the polynomial function g(x) whose graph passes through the points (0,4),(2,2), and (4,2); (b) Write the augmented matrix that you would use to find the coefficients of g(x). (c) Find the polynomial g(x).
To determine the polynomial function g(x),
(a) The system of equations: c = 4, 4a + 2b = -2, and 16a + 4b = -2.
(b) Augmented matrix: [0 0 1 | 4; 4 2 0 | -2; 16 4 0 | -2].
(c) Polynomial g(x) = -x^2 + 2x + 4 passing through (0,4), (2,2), and (4,2).
(a) To determine the polynomial function g(x) whose graph passes through the points (0, 4), (2, 2), and (4, 2), we can set up a system of linear equations.
Let's assume the polynomial function g(x) is of degree 2, so g(x) = ax^2 + bx + c.
Using the given points, we can substitute the x and y values to form the following equations:
Equation 1: g(0) = 4Substituting x = 0 and y = 4:
a(0)^2 + b(0) + c = 4
c = 4
Equation 2: g(2) = 2Substituting x = 2 and y = 2:
a(2)^2 + b(2) + c = 2
4a + 2b + 4 = 2
4a + 2b = -2
Equation 3: g(4) = 2Substituting x = 4 and y = 2:
a(4)^2 + b(4) + c = 2
16a + 4b + 4 = 2
16a + 4b = -2
Now we have a system of linear equations:
c = 4
4a + 2b = -2
16a + 4b = -2
(b) To find the coefficients of g(x), we can write the system of equations in augmented matrix form:
[0 0 1 | 4]
[4 2 0 | -2]
[16 4 0 | -2]
(c) To find the polynomial g(x), we need to solve the augmented matrix. Applying row operations to put the matrix in the reduced row-echelon form:
[1 0 0 | -1]
[0 1 0 | 2]
[0 0 1 | 4]
Therefore, the polynomial g(x) is g(x) = -x^2 + 2x + 4.
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Find g[f(1)]. f(x)=x^2−1;g(x)=2x−1
First, we find f(1) by substituting x = 1 into the function f(x) = x^2 - 1. f(1) = (1)^2 - 1 = 0. Next, we substitute f(1) = 0 into the function g(x) = 2x - 1. g[f(1)] = g(0) = 2(0) - 1 = -1.
The composition of functions is a mathematical operation where the output of one function is used as the input for another function. In this case, we have two functions, f(x) = x^2 - 1 and g(x) = 2x - 1. To find g[f(1)], we first evaluate f(1) by substituting x = 1 into f(x), resulting in f(1) = 0. Then, we substitute f(1) = 0 into g(x), which gives us g[f(1)] = g(0) = -1.
Therefore, g[f(1)] is equal to -1.
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\( f^{\prime}(x)=6+6 e^{x}+\frac{10}{x} ; \quad(1,7+6 e) \) \( f(x)= \)
\( f(x) = 6x + 6e^x + 10\ln|x| + C \), where \( C \) is the constant of integration.
To find \( f(x) \) from \( f'(x) \), we integrate \( f'(x) \) with respect to \( x \).
The integral of \( 6 \) with respect to \( x \) is \( 6x \).
The integral of \( 6e^x \) with respect to \( x \) is \( 6e^x \).
The integral of \( \frac{10}{x} \) with respect to \( x \) is \( 10\ln|x| \) (using the property of logarithms).
Adding these results together, we have \( f(x) = 6x + 6e^x + 10\ln|x| + C \), where \( C \) is the constant of integration.
Given the point \((1, 7 + 6e)\), we can substitute the values into the equation and solve for \( C \):
\( 7 + 6e = 6(1) + 6e^1 + 10\ln|1| + C \)
\( 7 + 6e = 6 + 6e + 10(0) + C \)
\( C = 7 \)
Therefore, the function \( f(x) \) is \( f(x) = 6x + 6e^x + 10\ln|x| + 7 \).
The function \( f(x) \) is a combination of linear, exponential, and logarithmic terms. The given derivative \( f'(x) \) was integrated to find the original function \( f(x) \), and the constant of integration was determined by substituting the given point \((1, 7 + 6e)\) into the equation.
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\( \int_{-2}^{3} x(x+2) d x \)
The value of the given integral [tex]\( \int_{-2}^{3} x(x+2) d x \)[/tex] is[tex]$$\int_{-2}^{3} x(x+2) d x = \int_{-2}^{3} (x^2+2x) d x = 11 + 25 = \boxed{36}$$[/tex] Thus, the answer is 36.
The integral can be solved using the distributive property and the power rule of integration. We start by expanding the integrand as follows:[tex]$$\int_{-2}^{3} x(x+2) d x = \int_{-2}^{3} (x^2+2x) d x$$[/tex]
Using the power rule of integration, we can integrate the integrand term by term. Applying the power rule of integration to the first term, we get[tex]$$\int_{-2}^{3} x^2 d x = \frac{x^3}{3}\bigg|_{-2}^{3} = \frac{3^3}{3} - \frac{(-2)^3}{3} = 11$$[/tex]
Applying the power rule of integration to the second term, we get[tex]$$\int_{-2}^{3} 2x d x = x^2\bigg|_{-2}^{3} = 3^2 - (-2)^2 = 5^2 = 25$$[/tex]
Therefore, the value of the given integral is[tex]$$\int_{-2}^{3} x(x+2) d x = \int_{-2}^{3} (x^2+2x) d x = 11 + 25 = \boxed{36}$$[/tex]
Thus, the answer is 36.
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Solve the equation. Check your answers. |x-3|=9
To solve the equation |x-3|=9, we consider two cases: (x-3) = 9 and -(x-3) = 9. In the first case, we find that x = 12. In the second case, x = -6. To check our answers, we substitute them back into the original equation, and they satisfy the equation. Therefore, the solutions to the equation are x = 12 and x = -6.
To solve the equation |x-3|=9, we need to consider two cases:
Case 1: (x-3) = 9
In this case, we add 3 to both sides to isolate x:
x = 9 + 3 = 12
Case 2: -(x-3) = 9
Here, we start by multiplying both sides by -1 to get rid of the negative sign:
x - 3 = -9
Then, we add 3 to both sides:
x = -9 + 3 = -6
So, the two solutions to the equation |x-3|=9 are x = 12 and x = -6.
The equation |x-3|=9 means that the absolute value of (x-3) is equal to 9. The absolute value of a number is its distance from zero on a number line, so it is always non-negative.
In Case 1, we consider the scenario where the expression (x-3) inside the absolute value bars is positive. By setting (x-3) equal to 9, we find one solution: x = 12.
In Case 2, we consider the scenario where (x-3) is negative. By negating the expression and setting it equal to 9, we find the other solution: x = -6.
To check our answers, we substitute x = 12 and x = -6 back into the original equation. For both cases, we find that |x-3| is indeed equal to 9. Therefore, our solutions are correct.
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The point (8t,2t+7) is on the graph of the function f(x) , and
the point (8t,−9t+9) is on the graph of the function g(x) . Find
the value of f⋅g at 8t .
The value of f⋅g at 8t is 9t² - 7t - 63. This result is obtained by substituting 8t into the functions f(x) and g(x) and multiplying the corresponding values. Therefore, the product of f(x) and g(x) evaluated at 8t yields the expression 9t² - 7t - 63.
To find the value of f⋅g at 8t, we need to multiply the values of f(x) and g(x) at 8t. Given that the point (8t, 2t + 7) lies on the graph of f(x) and the point (8t, -9t + 9) lies on the graph of g(x), we can substitute 8t into the respective functions.
For f(x), substituting 8t, we get f(8t) = 2(8t) + 7 = 16t + 7.
For g(x), substituting 8t, we get g(8t) = -9(8t) + 9 = -72t + 9.
To find the value of f⋅g at 8t, we multiply these two values:
f(8t) * g(8t) = (16t + 7) * (-72t + 9) = -1152t² + 144t - 504t - 63 = -1152t² - 360t - 63 = 9t² - 7t - 63.
Therefore, the value of f⋅g at 8t is 9t² - 7t - 63.
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Use I = Prt for simple interest to find the indicated quantity (use 360 days in a year): 1 = $750, r = 6%, t = 6 months, find P Use 1 = Prt for simple interest to find the indicated quantity (use 360 days in a year): P = $13500, t = 4 months, I = $517.50, find
1. The principal (P) is $625.
2. The interest rate (r) is 4%.
1. Given the formula for simple interest: I = Prt, we can rearrange it to solve for the principal (P): P = I / (rt).
For the first problem, we have:
I = $750
r = 6% (or 0.06)
t = 6 months (or 6/12 = 0.5 years)
Substituting these values into the formula, we get:
P = $750 / (0.06 * 0.5)
P = $750 / 0.03
P = $25,000 / 3
P ≈ $625
Therefore, the principal (P) is approximately $625.
2. For the second problem, we are given:
P = $13,500
t = 4 months (or 4/12 = 1/3 years)
I = $517.50
Using the same formula, we can solve for the interest rate (r):
r = I / (Pt)
r = $517.50 / ($13,500 * 1/3)
r = $517.50 / ($4,500)
r = 0.115 or 11.5%
Therefore, the interest rate (r) is 11.5%.
Note: It's important to pay attention to the units of time (months or years) and adjust them accordingly when using the simple interest formula. In the first problem, we converted 6 months to 0.5 years, and in the second problem, we converted 4 months to 1/3 years to ensure consistent calculations.
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how many sets of four consecutive positive integers are there such that the product of the four integers is less than 100,000?
There are 20 sets of four consecutive positive integers such that the product of the four integers is less than 100,000. The maximum value of the smallest integer in each set is 20.
To determine the number of sets of four consecutive positive integers whose product is less than 100,000, we can set up an equation and solve it.
Let's assume the smallest integer in the set is n. The four consecutive positive integers would be n, n+1, n+2, and n+3.
The product of these four integers is:
n * (n+1) * (n+2) * (n+3)
To count the number of sets, we need to find the maximum value of n that satisfies the condition where the product is less than 100,000.
Setting up the inequality:
n * (n+1) * (n+2) * (n+3) < 100,000
Now we can solve this inequality to find the maximum value of n.
By trial and error or using numerical methods, we find that the largest value of n that satisfies the inequality is n = 20.
Therefore, there are 20 sets of four consecutive positive integers whose product is less than 100,000.
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−2(z−2)≤16 or 13+z<22 Step 3 of 4: Using your anwwers from the previous steps, solve the overall inequality problem and express your anower in interval notation Use decimal form for mumerical qalues.
The overall inequality is z ≥ -6 or z < 9. The solution set can be expressed in interval notation as:(-∞, 9)U[-6, ∞)
Given: −2(z−2)≤16 or 13+z<22
We can use the following steps to solve the above-mentioned inequality problem:
Simplify each inequality
−2(z−2)≤16 or 13+z<22−2z + 4 ≤ 16 or z < 9
Solve for z in each inequality−2z ≤ 12 or z < 9z ≥ -6 or z < 9
Using your answers from the previous steps,
solve the overall inequality problem and express your answer in interval notation
Use decimal form for numerical values.
The overall inequality is z ≥ -6 or z < 9.
The solution set can be expressed in interval notation as:(-∞, 9)U[-6, ∞)
Thus, the solution to the given inequality is z ≥ -6 or z < 9 and it can be represented in interval notation as (-∞, 9)U[-6, ∞).
Thus, we can conclude that the solution to the given inequality is z ≥ -6 or z < 9. It can be represented in interval notation as (-∞, 9)U[-6, ∞).
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A sticker costs d cents. a marble costs 5 times as much. michael paid $13 for 6 such stickers and a few marbles. express the price of each marble in terms of d.
We are given that a marble costs 5 times as much as a sticker. The price of each marble in terms of d is 5d cents.
To express the price of each marble in terms of d, we first need to determine the cost of the stickers.
We know that Michael paid $13 for 6 stickers.
Since each sticker costs d cents, the total cost of the stickers can be calculated as [tex]6 * d = 6d[/tex] cents.
Next, we need to find the cost of the marbles.
We are given that a marble costs 5 times as much as a sticker.
Therefore, the cost of each marble can be expressed as 5 * d = 5d cents.
So, the price of each marble in terms of d is 5d cents.
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Compute and sketch the vector assigned to the points P=(0,−6,9) and Q=(8,1,0) by the vector field F=⟨xy,z 2
,x⟩. F(P)=
F(Q)=
The vector assigned to the point `P` is `<0,81,0>` and the vector assigned to the point `Q` is `<8,0,8>`.
We are required to compute and sketch the vector assigned to the points
`P=(0,−6,9)` and `Q=(8,1,0)` by the vector field `F=⟨xy,z^2,x⟩`.
Let's begin by computing the vector assigned to the point `
P=(0,−6,9)` by the vector field `F=⟨xy,z^2,x⟩`.
The value of `F(P)` can be computed as follows:`F(P) = <0*(-6),(9)^2,0>``F(P) = <0,81,0>`
Therefore, the vector assigned to the point `P=(0,−6,9)` by the vector field `F=⟨xy,z^2,x⟩` is `<0,81,0>`.
Next, we need to compute the vector assigned to the point `Q=(8,1,0)` by the vector field `F=⟨xy,z^2,x⟩`.
The value of `F(Q)` can be computed as follows:`F(Q) = <8*1,(0)^2,8>``F(Q) = <8,0,8>`
Therefore, the vector assigned to the point `Q=(8,1,0)` by the vector field `F=⟨xy,z^2,x⟩` is `<8,0,8>`.
Now, let's sketch the vectors assigned to the points `P` and `Q`.
The vector assigned to the point `P` is `<0,81,0>` and the vector assigned to the point `Q` is `<8,0,8>`.
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the distribution of home prices in salt lake city is skewed to the left. the median price is $150,000. specify the general location of the mean. a. lower than $150,000 b. higher than $150,000 c. it may fall anywhere to $150,000 d. equal to $150,000
The distribution of home prices in salt lake city is skewed to the left. the median price is $150,000. specify the general location of the mean a. lower than $150,000
In a left-skewed distribution, the mean is typically lower than the median. This is because the skewed tail on the left side pulls the mean in that direction. Since the median price in Salt Lake City is $150,000 and the distribution is skewed to the left, the general location of the mean would be lower than $150,000. Therefore, option a is the correct answer.
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Write the point-slope form of the line satisfying the given conditions. Then use the point-slope form of the equation to write the slope-intercept form of the equation Slope =8, passing through (−4,4) Type the point-slope form of the equation of the line. (Simplify your answer. Use integers or fractions for any numbers in the equation.)
The point-slope form of the equation is: y - 4 = 8(x + 4), which simplifies to the slope-intercept form: y = 8x + 36.
The point-slope form of a linear equation is given by y - y₁ = m(x - x₁), where (x₁, y₁) represents a point on the line and m represents the slope of the line.
Using the given information, the point-slope form of the equation of the line with a slope of 8 and passing through the point (-4, 4) can be written as:
y - 4 = 8(x - (-4))
Simplifying the equation:
y - 4 = 8(x + 4)
Expanding the expression:
y - 4 = 8x + 32
To convert the equation to slope-intercept form (y = mx + b), we isolate the y-term:
y = 8x + 32 + 4
y = 8x + 36
Therefore, the slope-intercept form of the equation is y = 8x + 36.
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a greeting card is 6 inches wide and 8 inches tall. point a is 3 inches from the fold, as shown. as the card is opened to an angle of 45 degrees, through how many more inches than point a does point b travel? express your answer as a common fraction in terms of $\pi$.
How many more inches point B travels than point A as the card is opened to an angle of 45 degrees, we need to calculate the arc length between point A and point B along the curved edge of the card. Point B travels π inches more than point A.
The curved edge of the card forms a quarter of a circle, since the card is opened to an angle of 45 degrees, which is one-fourth of a full 90-degree angle.
The radius of the circle is the height of the card, which is 8 inches. Therefore, the circumference of the quarter circle is one-fourth of the circumference of a full circle, which is given by 2πr, where r is the radius. The circumference of the quarter circle is (1/4) * 2π * 8 = 4π inches. Since point A is 3 inches from the fold, it travels an arc length of 3 inches.
To find how many more inches point B travels than point A, we subtract the arc length of point A from the arc length of the quarter circle:
4π - 3 = π inches.
Therefore, point B travels π inches more than point A.
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Write the interval notation for a set of all real numbers that are greater than 2 and less than or equal to 8. i have to show work too
The interval notation for a set of all real numbers that are greater than 2 and less than or equal to 8 can be written as (2, 8].
To explain how we arrived at this notation, let's break it down:
The symbol ( represents an open interval, meaning that the endpoint is not included in the set. In this case, since the numbers need to be greater than 2, we use (2 to indicate that 2 is excluded.
The symbol ] represents a closed interval, meaning that the endpoint is included in the set. In this case, since the numbers need to be less than or equal to 8, we use 8] to indicate that 8 is included.
Combining these symbols, we get (2, 8] as the interval notation for the set of real numbers that are greater than 2 and less than or equal to 8.
Remember, the notation (2, 8] means that the set includes all numbers between 2 (excluding 2) and 8 (including 8).
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Simplify the expression 4(3x−7)−5(2x−6) A) 2x−58 (B) 22x−58 (C) 2x+2 d)−2x−2
The simplified form of the given expression is `2x + 2` (option (C)
An expression contains one or more numbers and variables along with arithmetic operations.
Given expression: `4(3x−7)−5(2x−6)
`To simplify the given expression, we can follow the steps below
1. Apply distributive property for the coefficient `4` and `5` into the expression to remove the brackets`
12x - 28 - 10x + 30`
2. On combining like terms
`2x + 2`
Therefore, the simplified form of the given expression is `2x + 2`.
Hence, option (C) 2x + 2 is the correct answer.
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Find the value of each variable
15. [2 x 0]=[y 4 0]
16. [x + 261y - 3]= [-561 -4]
17. [1-247 - 32z + 4] = [1y -52x -47 -33z - 1]
18. [x21x + 2y]=[521 - 3]
19. [x+y 1] = [2 1]
[0 x-y] [0 8]
20. [y 21 x + y]=[x + 2218]
The solution for this system of equations is x = -1134 and y = 1080.To find the value of each variable in the given equations, we'll equate the corresponding elements on both sides.
[2x 0] = [y 4 0], Equating the elements: 2x = y, 0 = 4. Since the second equation, 0 = 4, is not true, there is no solution for this system of equations. [x + 261y - 3] = [-561 -4]. Equating the elements: x + 261y = -561
-3 = -4. Again, the second equation, -3 = -4, is not true. Therefore, there is no solution for this system of equations. [1-247 - 32z + 4] = [1y -52x -47 -33z - 1]. Equating the elements: 1 - 247 = 1-32z + 4 = y-52x - 47 = -33z - 1
The first equation simplifies to 1 - 247 = 1, which is not true. Thus, there is no solution for this system of equations. [x 21x + 2y] = [521 - 3]
Equating the elements:x = 5, 21x + 2y = 21, From the first equation, x = 5. Substituting x = 5 into the second equation: 21(5) + 2y = 21, 2y = -84, y = -42. The solution for this system of equations is x = 5 and y = -42. [x+y 1] = [2 1]. Equating the elements: x + y = 2, 1 = 1. The second equation, 1 = 1, is true for all values. From the first equation, we can't determine the exact values of x and y. There are infinitely many solutions for this system of equations. [0 x-y] = [0 8], Equating the elements:0 = 0, x - y = 8. The first equation is true for all values. From the second equation, we can't determine the exact values of x and y.
There are infinitely many solutions for this system of equations. [y 21 x + y] = [x + 2218]. Equating the elements: y = x + 2218, 21(x + y) = x. Simplifying the second equation: 21x + 21y = x, Rearranging the terms:
21x - x = -21y, 20x = -21y, x = (-21/20)y. Substituting x = (-21/20)y into the first equation: y = (-21/20)y + 2218. Multiplying through by 20 to eliminate the fraction: 20y = -21y + 44360, 41y = 44360, y = 1080. Substituting y = 1080 into x = (-21/20)y: x = (-21/20)(1080), x = -1134. The solution for this system of equations is x = -1134 and y = 1080.
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Consider points A(4,−1,3),B(3,1,7), and C(1,−3,−3). (a) Find the area of parallelogram ABCD with adjacent sides AB
and AC
. (b) Find the area of triangle ABC. (c) Find the shortest distance from point A to line BC.
(a) The area of the parallelogram ABCD is 4√17 square units.
(b) The area of triangle ABC is 2√17 square units.
(c) The shortest distance from A to line BC is frac{30\sqrt{170}}{13} units.
Given points A(4,−1,3),B(3,1,7), and C(1,−3,−3).
(a) Find the area of parallelogram ABCD with adjacent sides AB and AC
.The formula for the area of the parallelogram in terms of sides is:
\text{Area} = |\vec{a} \times \vec{b}| where a and b are the adjacent sides of the parallelogram.
AB = \vec{b} and AC = \vec{a}
So,\vec{a} = \begin{bmatrix} 1 - 4 \\ -3 + 1 \\ -3 - 3 \end{bmatrix} = \begin{bmatrix} -3 \\ -2 \\ -6 \end{bmatrix} and
\vec{b} = \begin{bmatrix} 3 - 4 \\ 1 + 1 \\ 7 - 3 \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \\ 4 \end{bmatrix}
Now, calculating the cross product of these vectors, we have:
\begin{aligned} \vec{a} \times \vec{b} &= \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ -3 & -2 & -6 \\ -1 & 2 & 4 \end{vmatrix} \\ &= \begin{bmatrix} 2\vec{i} - 24\vec{j} + 8\vec{k} \end{bmatrix} \end{aligned}
The area of the parallelogram ABCD = |2i − 24j + 8k| = √(2²+24²+8²) = 4√17 square units.
(b) Find the area of triangle ABC.
The formula for the area of the triangle in terms of sides is:
\text{Area} = \dfrac{1}{2} |\vec{a} \times \vec{b}| where a and b are the two sides of the triangle which are forming a vertex.
Let AB be a side of the triangle.
So, vector \vec{a} is same as vector \vec{AC}.
Therefore,\vec{a} = \begin{bmatrix} 1 - 4 \\ -3 + 1 \\ -3 - 3 \end{bmatrix} = \begin{bmatrix} -3 \\ -2 \\ -6 \end{bmatrix} and \vec{b} = \begin{bmatrix} 3 - 4 \\ 1 + 1 \\ 7 - 3 \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \\ 4 \end{bmatrix}
Now, calculating the cross product of these vectors, we have:
\begin{aligned} \vec{a} \times \vec{b} &= \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ -3 & -2 & -6 \\ -1 & 2 & 4 \end{vmatrix} \\ &= \begin{bmatrix} 2\vec{i} - 24\vec{j} + 8\vec{k} \end{bmatrix} \end{aligned}
The area of the triangle ABC is:$$\begin{aligned} \text{Area} &= \dfrac{1}{2} |\vec{a} \times \vec{b}| \\ &= \dfrac{1}{2} \cdot 4\sqrt{17} \\ &= 2\sqrt{17} \end{aligned}$$
(c) Find the shortest distance from point A to line BC.
Let D be the foot of perpendicular from A to the line BC.
Let \vec{v} be the direction vector of BC, then the vector \vec{AD} will be perpendicular to the vector \vec{v}.
The direction vector \vec{v} of BC is:
\vec{v} = \begin{bmatrix} 1 - 3 \\ -3 - 1 \\ -3 - 7 \end{bmatrix} = \begin{bmatrix} -2 \\ -4 \\ -10 \end{bmatrix} = 2\begin{bmatrix} 1 \\ 2 \\ 5 \end{bmatrix}
Therefore, the vector \vec{v} is collinear to the vector \begin{bmatrix} 1 \\ 2 \\ 5 \end{bmatrix} and hence we can take \vec{v} = \begin{bmatrix} 1 \\ 2 \\ 5 \end{bmatrix}, which will make the calculations easier.
Let the point D be (x,y,z).
Then the vector \vec{AD} is:\vec{AD} = \begin{bmatrix} x - 4 \\ y + 1 \\ z - 3 \end{bmatrix}
As \vec{AD} is perpendicular to \vec{v}, the dot product of \vec{AD} and \vec{v} will be zero:
\begin{aligned} \vec{AD} \cdot \vec{v} &= 0 \\ \begin{bmatrix} x - 4 & y + 1 & z - 3 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 2 \\ 5 \end{bmatrix} &= 0 \\ (x - 4) + 2(y + 1) + 5(z - 3) &= 0 \end{aligned}
Simplifying, we get:x + 2y + 5z - 23 = 0
This equation represents the plane which is perpendicular to the line BC and passes through A.
Now, let's find the intersection of this plane and the line BC.
Substituting x = 3t + 1, y = -3t - 2, z = -3t - 3 in the above equation, we get:
\begin{aligned} x + 2y + 5z - 23 &= 0 \\ (3t + 1) + 2(-3t - 2) + 5(-3t - 3) - 23 &= 0 \\ -13t - 20 &= 0 \\ t &= -\dfrac{20}{13} \end{aligned}
So, the point D is:
\begin{aligned} x &= 3t + 1 = -\dfrac{41}{13} \\ y &= -3t - 2 = \dfrac{46}{13} \\ z &= -3t - 3 = \dfrac{61}{13} \end{aligned}
Therefore, the shortest distance from A to the line BC is the distance between points A and D which is:
\begin{aligned} \text{Distance} &= \sqrt{(4 - (-41/13))^2 + (-1 - 46/13)^2 + (3 - 61/13)^2} \\ &= \dfrac{30\sqrt{170}}{13} \end{aligned}
Therefore, the shortest distance from point A to line BC is \dfrac{30\sqrt{170}}{13}.
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a) use the product rule to find the derivative of the given function. b) find the derivative by multiplying the expressions first. y=x^4*x^6
The derivative of y = x^4 * x^6 using the product rule is y' = 4x^3 * x^6 + x^4 * 6x^5.
To find the derivative of the function y = x^4 * x^6, we can use the product rule, which states that the derivative of the product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function.
Applying the product rule to y = x^4 * x^6, we have:
y' = (x^4)' * (x^6) + (x^4) * (x^6)'
Differentiating x^4 with respect to x gives us (x^4)' = 4x^3, and differentiating x^6 with respect to x gives us (x^6)' = 6x^5.
Substituting these derivatives into the product rule, we get:
y' = 4x^3 * x^6 + x^4 * 6x^5.
Simplifying this expression, we have:
y' = 4x^9 + 6x^9 = 10x^9.
Therefore, the derivative of y = x^4 * x^6 is y' = 10x^9.
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Suppose that f(x) is a function for which f(2)=10, the derwative f'(2)=0, and the second decivative f "(2)=−4. Which stitement best describes f(x) at the point x=2?.a. f(x) has a lecal minimum value at x=2. b.f(x) does net have a local extreme value at x=2 c.f(x) thas a keal maximum value at x=2 d.f(x) hat an intlection point at x=2
The derivative is zero and the second derivative is negative, which means that the function has a point of inflection. Therefore, the best statement that describes f(x) at x = 2 is f(x) does not have a local extreme value at x = 2. And f(x) has an inflection point at x = 2.
Given, f(2) = 10, f'(2) = 0, and f''(2) = -4We need to find the statement that describes f(x) at x = 2.The first derivative of a function f(x) gives the slope of the function at any point. The second derivative gives the information about the curvature of the function. Let's check the options:
a) f(x) has a local minimum value at x = 2.
We can say that this option is incorrect as the derivative of the function is zero at x = 2, which indicates that the function does not change at x = 2.
b) f(x) does not have a local extreme value at x = 2.
This option is correct as the derivative is zero and the second derivative is negative, which means that the function has a point of inflection.
c) f(x) has a local maximum value at x = 2. This option is incorrect as the sign of the second derivative indicates that the point x = 2 is a point of inflection rather than a maximum or a minimum.d) f(x) has an inflection point at x = 2. This option is correct as the second derivative of the function is negative, indicating a point of inflection.
Therefore, the best statement that describes f(x) at x = 2 is f(x) does not have a local extreme value at x = 2. And f(x) has an inflection point at x = 2.
We can say that this option is incorrect as the derivative of the function is zero at x = 2, which indicates that the function does not change at x = 2.
This option is correct as the derivative is zero and the second derivative is negative, which means that the function has a point of inflection.
This option is incorrect as the sign of the second derivative indicates that the point x = 2 is a point of inflection rather than a maximum or a minimum. This option is correct as the second derivative of the function is negative, indicating a point of inflection. Therefore, the best statement that describes f(x) at x = 2 is f(x) does not have a local extreme value at x = 2. And f(x) has an inflection point at x = 2.
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2. (13pt) The following complex numbers are giving: z 1
=−2−2j,z 2
=− 3
+j&z 3
=a+bj where a∈R,b∈R (a) (3pt) If ∣z 1
z 3
∣=16, find the modulus z 3
. (b) (3pt) Given further that: arg( z 2
z 3
)= 12
7π
determine the argument z 3
. (c) (7pt) Find the values of a and b, and hence find z 1
z 3
.
The complex numbers is:
(a) |z3| = 4√2
(b) arg(z3) = -13π/42
(c) a = -2, b = -1, z1z3 = 6 + 6j
(a) If |z₁z₃| = 16, we know that |z₁z₃| = |z₁| * |z₃|. Since |z₁| = √((-2)² + (-2)²) = √8 = 2√2, we can write the equation as 2√2 * |z₃| = 16. Solving for |z3|, we get |z₃| = 16 / (2√2) = 8 / √2 = 4√2.
(b) Given arg(z₂z₃) = 12π/7, we can write arg(z₂z₃) = arg(z₂) - arg(z₃). The argument of z₂ is arg(z₂) = arg(-3 + j) = arctan(1/(-3)) = -π/6. Therefore, we have -π/6 - arg(z₃) = 12π/7. Solving for arg(z₃), we get arg(z₃) = -π/6 - 12π/7 = -13π/42.
(c) To find the values of a and b, we equate the real and imaginary parts of z₃ to a and b respectively. From z₃ = a + bj, we have Re(z₃) = a and Im(z₃) = b. Since Re(z₃) = -2 and Im(z₃) = -1, we can conclude that a = -2 and b = -1.
Now, to find z₁z₃, we multiply z₁ and z₃:
z₁z₃ = (-2 - 2j)(-2 - j) = (-2)(-2) - (-2)(j) - (-2)(2j) - (j)(2j) = 4 + 2j + 4j - 2j^2 = 4 + 6j - 2(-1) = 6 + 6j.
Therefore, z₁z₃ = 6 + 6j.
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Solve each inequality. (Lesson 0-6) -14 n ≥ 42
To solve the inequality [tex]-14n ≥ 42[/tex], we need to isolate the variable n. Now, we know that the solution to the inequality [tex]-14n ≥ 42[/tex] is [tex]n ≤ -3.[/tex]
To solve the inequality -14n ≥ 42, we need to isolate the variable n.
First, divide both sides of the inequality by -14.
Remember, when dividing or multiplying both sides of an inequality by a negative number, you need to reverse the inequality symbol.
So, [tex]-14n / -14 ≤ 42 / -14[/tex]
Simplifying this, we get n ≤ -3.
Therefore, the solution to the inequality -14n ≥ 42 is n ≤ -3.
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Since 56 is greater than or equal to 42, the inequality is true.
To solve the inequality -14n ≥ 42, we need to isolate the variable n.
First, let's divide both sides of the inequality by -14. Remember, when dividing or multiplying an inequality by a negative number, we need to reverse the inequality symbol.
-14n ≥ 42
Divide both sides by -14:
n ≤ -3
So the solution to the inequality -14n ≥ 42 is n ≤ -3.
This means that any value of n that is less than or equal to -3 will satisfy the inequality. To verify this, you can substitute a value less than or equal to -3 into the original inequality and see if it holds true. For example, if we substitute -4 for n, we get:
-14(-4) ≥ 42
56 ≥ 42
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how many ways are there to select 9 players for the starting lineup and a batting order for the 9 starters? g
There are 362,880 ways to select 9 players for the starting lineup and a batting order for the 9 starters based on the concept of combinations.
To calculate the number of ways to select 9 players for the starting lineup, we need to consider the combination formula. We have to choose 9 players from a pool of players, and order does not matter. The combination formula is given by:
[tex]C(n, r) =\frac{n!}{(r!(n - r)!}[/tex]
Where n is the total number of players and r is the number of players we need to select. In this case, n = total number of players available and r = 9.
Assuming there are 15 players available, we can calculate the number of ways to select 9 players:
[tex]C(15, 9) = \frac{15!}{9!(15 - 9)!} = \frac{15!}{9!6!}[/tex]
To determine the batting order, we need to consider the permutations of the 9 selected players. The permutation formula is given by:
P(n) = n!
Where n is the number of players in the batting order. In this case, n = 9.
P(9) = 9!
Now, to calculate the total number of ways to select 9 players for the starting lineup and a batting order, we multiply the combinations and permutations:
Total ways = C(15, 9) * P(9)
= (15! / (9!6!)) * 9!
After simplification, we get:
Total ways = 362,880
There are 362,880 ways to select 9 players for the starting lineup and a batting order for the 9 starters. This calculation takes into account the combination of selecting 9 players from a pool of 15 and the permutation of arranging the 9 selected players in the batting order.
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Determine the interval of convergence for the power series that can be used to represent the function f(x)=− 3/18x+4 , centered at x=0. Write your answer in interval notation.
The interval of convergence for the power series representing the function f(x) = -3/18x+4, centered at x=0, is (-6, 2).
To determine the interval of convergence for the power series, we can use the ratio test. The ratio test states that if we have a power series ∑(n=0 to ∞) cₙ(x-a)ⁿ, and we calculate the limit of the absolute value of the ratio of consecutive terms as n approaches infinity, if the limit is L, then the series converges if L < 1 and diverges if L > 1.
In this case, the given function is f(x) = -3/18x+4. We can rewrite this as f(x) = -1/6 * (1/x - 4). Now, we can compare this with the form of a power series, where a = 0. Taking the ratio of consecutive terms, we have cₙ(x-a)ⁿ / cₙ₊₁(x-a)ⁿ⁺¹ = (1/x - 4) / (1/x - 4) * (x-a) = 1 / (x-a).
Taking the limit as n approaches infinity, we find that the limit of the absolute value of the ratio is 1/|x|. For the series to converge, this limit must be less than 1, so we have 1/|x| < 1. Solving this inequality, we get |x| > 1, which implies -∞ < x < -1 or 1 < x < ∞.
However, we need to consider the interval centered at x=0. From the derived intervals, we can see that the interval of convergence is (-1, 1). But since the series is centered at x=0, we need to expand the interval symmetrically around x=0. Hence, the final interval of convergence is (-1, 1).
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How could they calculate the present value of a stock given that it presents different growths (g) during their analysis? In the first 5 years it grows by 5% and from then on it grows by 3% since there is a rate of r 5% to discount flows and the initial share price is $1.00
To calculate the present value of a stock with varying growth rates, you can use the discounted cash flow (DCF) method. In this case, where the stock grows by 5% for the first 5 years and then grows by 3% thereafter, and with a discount rate of 5%, the present value can be determined.
To calculate the present value, you would discount each future cash flow to its present value using the appropriate discount rate. In this scenario, you would calculate the present value for each year separately based on the corresponding growth rate. For the first 5 years, the growth rate is 5%. Let's assume the cash flow at the end of year 1 is X. The present value of this cash flow would be X / (1 + 0.05)¹, as it is discounted by the rate of 5%. Similarly, for year 2, the cash flow would be X * 1.05, and its present value would be X * 1.05 / (1 + 0.05)². This process is repeated for each of the first 5 years.
From the 6th year onwards, the growth rate is 3%. So, for year 6, the cash flow would be X * 1.05^5 * 1.03, and its present value would be X * 1.05^5 * 1.03 / (1 + 0.05)⁶. The same calculation is performed for subsequent years. By summing up the present values of each cash flow, you would obtain the present value of the stock. The initial share price of $1.00 would also be considered in the present value calculation, typically as the cash flow at year 0.
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If X and Y are independent random variables with variances σ2X = 5 and σ2Y = 3, find the variance of the random variable Z = −2X +4Y − 3.
X and Y are independent random variables with variances σ2X = 5 and σ2Y = 3, The variance of the random variable Z = −2X +4Y − 3 is 68.
To find the variance of the random variable Z = -2X + 4Y - 3, we need to apply the properties of variance and independence of random variables.
First, let's find the variance of -2X + 4Y:
Var(-2X + 4Y) = (-2)² × Var(X) + 4² × Var(Y)
Given that Var(X) = σ²X = 5 and Var(Y) = σ²Y = 3:
Var(-2X + 4Y) = 4 × 5 + 16 × 3 = 20 + 48 = 68
Now, let's find the variance of Z:
Var(Z) = Var(-2X + 4Y - 3)
Since the variance operator is linear, we can rewrite this as:
Var(Z) = Var(-2X + 4Y) + Var(-3)
Since Var(-3) is a constant, its variance is zero:
Var(-3) = 0
Therefore, we can simplify the equation:
Var(Z) = Var(-2X + 4Y) + 0 = Var(-2X + 4Y) = 68
Thus, the variance of the random variable Z is 68.
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