Answer:
10,000 * (square miles in the city)
Step-by-step explanation:
You've been given a population of people for every square mile (10,000) and you're trying to figure out how many people live in the city.
To convert people per a square mile into people, you'll need to know how many square miles you have. I assume you've been given this as the size of the city or how many square miles the city takes up. Just multiply the amount of people per a square mile by the amount of square miles in the city, and it'll give you the population of people in every square mile of the city (the population).
Note:
This is people per a square mile, or in other words, for every square mile of city, 10,000 people live there. If you have 2 square miles of city, then 2 * 10,000 = 20,000 people live there.
Please help divide 5 652 to the ratio 2:3:5 I will give brainliesters
thank you in advance
Answer:
105
Step-by-step explanation:
(-20+5)(58-65)
(-15)(-7)
105
The regular price of a camera is $500. The camera is on sale for 25% off the regular price. What is the sale price of the camera in dollars and cents?
Answer:
375
Step-by-step explanation:
500-125(25% of 500)=375
A sample of 18 male students was asked how much they spent on textbooks this semester. The sample variance was s2M = 35.05. A sample of eight female students was asked the same question, and the sample variance was s2F = 18.40. (Data collected by Megan Damron and Spencer Solomon, 2009.) Assume that the amount spent on textbooks is normally distributed for both the populations of male students and of female students.
a. Calculate a 90% confidence interval estimate for sigma2M, the population variance of the amount spent on textbooks by male students.
b. Calculate a 90% con?dence interval estimate for sigma2M, the population variance of the amount spent on textbooks by female students.
a.0.05 ≤ P (χ²(17) < (18 - 1)35.05/σ²M < χ²(0.95)(17))0.05 ≤ P (χ²(17) < 577.57/σ²M < χ²(0.95)(17))0.05 ≤ P (χ²(17) < 577.57/σ²M < 28.412)The above inequality represents the 90%
b. 0.05 ≤ P (χ²(7) < (8 - 1)18.40/σ²F < χ²(0.95)(7))0.05 ≤ P (χ²(7) < 122.18/σ²F < χ²(0.95)(7))0.05 ≤ P (χ²(7) < 122.18/σ²F < 14.067)
The above inequality represents the 90% confidence interval estimate for sigma2F.
a. 90% confidence interval estimate for sigma2M:We are given that the sample variance is s²M=35.05 and a sample of 18 male students was asked how much they spent on textbooks. We are also given that the amount spent on textbooks is normally distributed for both the populations of male students.
Using the Chi-Square distribution, we have: (n - 1)s²M/σ²M follows a Chi-Square distribution with n - 1 degrees of freedom.
Then, (n - 1)s²M/σ²M ~ χ²(n - 1)For a 90% confidence interval estimate, we can write: 0.05 ≤ P (χ²(17) < (n - 1)s²M/σ²M < χ²(0.95)(17))
Using the table of chi-square values with (n - 1) degrees of freedom, we have:χ²(0.05)(17) = 8.567χ²(0.95)(17) = 28.412
Substituting the values, we have:0.05 ≤ P (χ²(17) < (18 - 1)35.05/σ²M < χ²(0.95)(17))0.05 ≤ P (χ²(17) < 577.57/σ²M < χ²(0.95)(17))0.05 ≤ P (χ²(17) < 577.57/σ²M < 28.412)The above inequality represents the 90%
b. confidence interval estimate for sigma2M.b. 90% con? dence interval estimate for sigma2F:Using the same concept as above, we can write: 0.05 ≤ P (χ²(7) < (n - 1)s²F/σ²F < χ²(0.95)(7))
Using the table of chi-square values with (n - 1) degrees of freedom, we have:χ²(0.05)(7) = 3.357χ²(0.95)(7) = 14.067
Substituting the values, we have:0.05 ≤ P (χ²(7) < (8 - 1)18.40/σ²F < χ²(0.95)(7))0.05 ≤ P (χ²(7) < 122.18/σ²F < χ²(0.95)(7))0.05 ≤ P (χ²(7) < 122.18/σ²F < 14.067)
The above inequality represents the 90% confidence interval estimate for sigma2F.
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To calculate a 90% confidence interval estimate for σ2M, the population variance of the amount spent on textbooks by male students, we use the formula below:
[tex]$$\chi_{0.05,17}^2 <\frac{(n - 1)s^2}{\sigma^2} < \chi_{0.95,17}^2$$[/tex]
where n = 18, s2M = 35.05, df = n - 1 = 17, and χα2,
df is the critical value from the chi-squared distribution with df degrees of freedom.
We know that:
[tex]$$\chi_{0.05,17}^2 = 8.909$$[/tex]
and
[tex]$$\chi_{0.95,17}^2 = 31.410$$[/tex]
Substituting these values, we have:
[tex]$$8.909 < \frac{(18-1)(35.05)}{\sigma^2} < 31.410$$[/tex]
Solving for σ2, we have:
[tex]$$\frac{(18-1)(35.05)}{31.410} < \sigma^2 < \frac{(18-1)(35.05)}{8.909}$$[/tex]
Hence, a 90% confidence interval estimate for σ2M is:
[tex]$$(48.704, 194.154)$$b.[/tex]
To calculate a 90% confidence interval estimate for σ2F, the population variance of the amount spent on textbooks by female students, we use the formula below:
[tex]$$\chi_{0.05,7}^2 <\frac{(n - 1)s^2}{\sigma^2} < \chi_{0.95,7}^2$$[/tex]
where n = 8, s2F = 18.40, df = n - 1 = 7, and χα2,
df is the critical value from the chi-squared distribution with df degrees of freedom.
We know that:
[tex]$$\chi_{0.05,7}^2 = 14.067$$[/tex] and
[tex]$$\chi_{0.95,7}^2 = 2.998$$[/tex]
Substituting these values, we have:
[tex]$$14.067 < \frac{(8-1)(18.40)}{\sigma^2} < 2.998$$[/tex]
Solving for σ2, we have:
[tex]$$\frac{(8-1)(18.40)}{2.998} < \sigma^2 < \frac{(8-1)(18.40)}{14.067}$$[/tex]
Hence, a 90% confidence interval estimate for σ2F is:
[tex]$$(7.176, 23.622)$$[/tex]
Therefore, the 90% confidence interval estimate for σ2M,
the population variance of the amount spent on textbooks by male students, is (48.704, 194.154), while the 90% confidence interval estimate for σ2F,
the population variance of the amount spent on textbooks by female students, is (7.176, 23.622).
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reverse the order of integration and then evaluate the integral.
₀∫¹₄ᵧ∫⁴ x⁴eˣ^²ʸ dx dy
a. e¹⁶ - 1
b. e¹⁶ - 68/3
c. 4e¹⁶-68 / 3
d. 4e¹⁶ - 68
The correct answer is option (b): [tex]e^16 - 68/3[/tex]. The approximate value of this expression is [tex]e^16 - 68/3[/tex].
To reverse the order of integration, we need to change the order of integration and rewrite the limits of integration accordingly.
The given integral is:
∫₀¹₄ ᵧ ∫⁴ x⁴e^(x²ʸ) dx dy
To reverse the order of integration, we integrate with respect to y first. The limits of integration for y are 0 to 14ᵧ. The limits of integration for x will depend on the value of y.
∫₀¹₄ ᵧ ∫⁴ x⁴e^(x²ʸ) dx dy
Let's integrate with respect to x first:
∫⁴ x⁴e^(x²ʸ) dx = [1/5 x⁵e^(x²ʸ)]⁴₀
Now we can rewrite the integral with reversed order of integration:
∫₀¹₄ dy ∫⁴₀ x⁴e^(x²ʸ) dx
Plugging in the limits of integration for x:
∫₀¹₄ dy [1/5 x⁵e^(x²ʸ)]⁴₀
Now we can evaluate the integral:
∫₀¹₄ dy [1/5 (⁴)⁵e^(⁴²ʸ) - 1/5 (⁰)⁵e^(⁰²ʸ)]
Simplifying:
∫₀¹₄ dy [1/5 (1024e^(16ʸ) - 1)]
Now integrate with respect to y:
[1/5 (1024e^(16ʸ) - 1)]¹₄
Plugging in the limits of integration for y:
[1/5 (1024e^(1614) - 1)] - [1/5 (1024e^(160) - 1)]
Simplifying:
[1/5 (1024e^(224) - 1)] - [1/5 (1024e^(0) - 1)]
[1/5 (1024e^(224) - 1)] - [1/5 (1024 - 1)]
[1/5 (1024e^(224) - 1)] - [1/5 (1023)]
[1/5 (1024e^(224) - 1)] - [204.6]
To evaluate the expression, we need the actual numerical value for e^(224). Using a calculator, we find that e^(224) is an extremely large number. Therefore, we can approximate it as e^(224) ≈ 2.4858 x 10^97.
Plugging in the value:
[1/5 (1024 x (2.4858 x 10^97) - 1)] - [204.6]
Simplifying the expression:
[2.4858 x 10^97 - 1] / 5 - 204.6
The approximate value of this expression is:
e^16 - 68/3
Therefore, the correct answer is option (b): e^16 - 68/3.
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Write two expressions that are equivalent to 3 . 10^-6
Answer:
636372
Step-by-step explanation:
the answer is 636372
On a test that has a normal distribution, a score of 32 falls three standard
deviations below the mean, and a score of 82 falls two standard deviations
above the mean. Determine the mean of this test.
Answer:
Mean = 62
Step-by-step explanation:
X = mean ± sd
82 = mean + 2(s.d) - - (1)
32 = mean - 3sd ___(2)
82 = mean + 2sd
82 - 32= 2sd + 3sd
50 = 5sd
sd = 50/5
sd = 10
From (2):
32 = mean - 3sd
32 = mean - 3(10)
32 = mean - 30
32 + 30 = mean
62 = mean
The graph shows the amount of money in a savings account over a period of 10 weeks.
How much money was saved over the 10 week period?
A) $50
B) $80
C) $110
D) $112
Answer:
A) $50
Step-by-step explanation:
Answer:
+$50
Step-by-step explanation:
Day #0 = $30
Day #10 = $80
80 - 30 = 50
:P
What is a simpler form of
5n(3n³−n²+8) ?
Four times the sum of a number r and 4 is equal to 20 less than 10 times the number r. What is the value of r ?
Answer:
r = 6
Step-by-step explanation:
According to the question,
4(r + 4) = 10r - 20
4r + 16 = 10r - 20
16 + 20 = 10r - 4r
6r = 36
r = 36 / 6
r = 6
Can someone plz help me
Answer:
28yd^2
Step-by-step explanation:
the answer is 28 yards squared bud :)
Find the relative maximum and minimum values of f(x,y) = x3/3 + 2xy + y2 - 3x + 1. 3
The critical point (1, -1) represents a relative minimum of f(x, y) with a value of -1/3 and (1, -1) is the only extremum or relative maximum of the function.
To find the relative maximum and minimum values of the function f(x, y) = ([tex]x^3[/tex])/3 + 2xy +[tex]y^2[/tex] - 3x + 1, we need to analyze its critical points and classify them using the second partial derivative test.
To find the critical points, we need to compute the partial derivatives of f with respect to x and y and set them equal to zero:
∂f/∂x = [tex]x^2[/tex] + 2y - 3 = 0
∂f/∂y = 2x + 2y = 0
Solving these equations simultaneously, we find x = 1 and y = -1.
Thus, the critical point is (1, -1).
Next, we need to compute the second partial derivatives and evaluate them at the critical point:
∂²f/∂x² = 2
∂²f/∂y² = 2
∂²f/∂x∂y = 2
Now, we can use the second partial derivative test to classify the critical point.
The discriminant D = (∂²f/∂x²) × (∂²f/∂y²) - [tex]\left(\frac{{\partial^2 f}}{{\partial x \partial y}}\right)^2[/tex] = (2)(2) - [tex](2)^2[/tex] = 0.
Since D = 0, the test is inconclusive.
To determine the nature of the critical point, we can examine the function near the critical point.
Evaluating f at the critical point (1, -1), we find f(1, -1) = [tex](1^3)[/tex]/3 + 2(1)(-1) + [tex](-1)^2[/tex] - 3(1) + 1 = -1/3.
Hence, the critical point (1, -1) represents a relative minimum of f(x, y) with a value of -1/3.
There are no other critical points to consider, so we can conclude that (1, -1) is the only extremum of the function.
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evaluate the following expressions. your answer must be an exact angle in radians and in the interval [ 0 , π ] . example: enter pi/6 for π 6 .
The expression to evaluate is arccos([tex]\sqrt(3)[/tex]/2) - arcsin(1/2). The exact angle in radians in the interval [0, π] for this expression is π/6.
To evaluate the given expression, we start by calculating the values inside the trigonometric functions. The square root of 3 divided by 2 is equal to 0.866, and 1 divided by 2 is equal to 0.5. The arccos function gives us the angle whose cosine is equal to the input. In this case, the cosine of the angle we are looking for is[tex]\sqrt(3)[/tex]/2. Using the unit circle, we find that this angle is π/6 radians. Next, we calculate the arcsin of 1/2, which gives us the angle whose sine is equal to the input. This angle is π/6 radians as well. Finally, we subtract the two angles to get our result: π/6 - π/6 = 0. Therefore, the exact angle in radians in the interval [0, π] for the given expression is π/6.
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Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals.
f(x)= 9x
f(1/2) =
f(square root of 6)=
f(-2)=
f(0.4)=
Given function is: f(x)= 9xTo find the values of the given function at the indicated values: Round off the answer to 3 decimal places.1. f(1/2).
f(1/2):
Plug x = 1/2 into the function:
f(1/2) = 9(1/2)
Simplifying, we find that f(1/2) = 4.5.
Similarly, for the remaining parts,
Substitute x = 1/2 in the given function. f(1/2) = 9 (1/2) = 4.5002. f(√6).
Substitute x = √6 in the given function. f(√6) = 9 √6 = 27.7123. f(-2). Substitute x = -2 in the given function. f(-2) = 9 (-2) = -18.0004. f(0.4).
Substitute x = 0.4 in the given function. f(0.4) = 9 (0.4) = 3.600. The required values of the given function are: f(1/2) = 4.500f(√6) = 27.712f(-2) = -18.000f(0.4) = 3.600.
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Let w = {C5 %):a,ber } with the standard operations in M2.2- Which of the following statements is true? The 2x2 identity matrix is in W W is not a subspace of M2x2 because it does not contain the zero matrix the above is true W is a subspace of M2x2. the above is true None of the mentioned
The statement "W is a subspace of M2x2" is true because W satisfies the three conditions for being a subspace.
To determine whether W is a subspace of M2x2, we need to verify three conditions:
W is non-empty: Since W is defined as the set of all 2x2 matrices with a fixed entry of 5, it contains at least one matrix (such as [[5, 5], [5, 5]]).
W is closed under addition: For any two matrices A and B in W, their sum A + B will also have the fixed entry of 5 in the corresponding position. Therefore, the sum of any two matrices in W will still be in W.
W is closed under scalar multiplication: For any matrix A in W and any scalar c, the scalar multiple cA will also have the fixed entry of 5 in the corresponding position. Hence, the scalar multiple of any matrix in W will still be in W.
Since W satisfies all three conditions, it is indeed a subspace of M2x2.
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You estimate that a lamb weighs 10 pounds. The actual weight of the lamb is 13 pounds. Find the percent error. Round your answer to the nearest hundredth of a percent.
%
Which statement is true?
Answer:
C would be your answer!
Step-by-step explanation:
Need help in proving theorems of square please ASAP
Answer:
The length of [tex]\overline{JO}[/tex] is 48.
Step-by-step explanation:
A square is a quadrilateral whose four sides have the same length and four internal angles have the same measure. The sum of measures of internal angles in quadrilaterals equals 360°. Let [tex]m\,\angle BOJ = 4\cdot x -6[/tex] and [tex]BO = 2\cdot x - 8[/tex], the value of [tex]x[/tex] is:
[tex]4\cdot (4\cdot x - 6) = 360^{\circ}[/tex]
[tex]16\cdot x -24 = 360^{\circ}[/tex]
[tex]16\cdot x = 384^{\circ}[/tex]
[tex]x = 24[/tex]
And the length of [tex]JO[/tex] is:
[tex]JO = BO = 2\cdot x - 8[/tex]
[tex]JO = 40[/tex]
The length of [tex]\overline{JO}[/tex] is 48.
What does x equal please help
Answer: 5 1/2 (5.5)
Step-by-step explanation:
add 4 2/3 to 5/6
Find the function value, if possible.
f(x)= -2x -2 x<-1
x^2 +2x -1 x>=1
The function value for f(x) depends on the value of x. The function value of f(x) can be determined as follows:
- For x < -1, f(x) = -2x - 2.
- For x ≥ 1, [tex]f(x) = x^2 + 2x - 1[/tex].
The function value for f(x) depends on the value of x. If x is less than -1, then the function f(x) can be calculated as -2x - 2. On the other hand, if x is greater than or equal to 1, then f(x) can be determined as [tex]x^2 + 2x - 1[/tex].
To summarize, the function f(x) is defined differently based on the value of x. For x values less than -1, f(x) equals -2x - 2. For x values greater than or equal to 1, f(x) is given by [tex]x^2 + 2x - 1[/tex]
In the first paragraph, provided a brief summary of the function value based on the given conditions. In the second paragraph, explained how the function f(x) is defined for different ranges of x.
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A store purchases a shirt for $16.89. The store marks up the shirt by 20%. Right now, they are having a sale for 40% off any item. What is the sale price with a 7.25% tax?
Answer:
$13.04
Step-by-step explanation:
First, multiply 16.89 by 1.2, 0.6, and 1.0725. You should get a weird number on your calculator saying 13.042458, but just round 2 to the 4 ( 4 stays the same) and remove all the other numbers to get $13.04
The height of a parallelogram is 4 millimeters more than its base. If the area of the
parallelogram is 221 square millimeters, find its base and height.
Answer:
523 inches
Step-by-step explanation:
got it right on edg
How should the coefficients a, b, and c be chosen so that the system ax+by - 32-3 - 2x - by+cz= -1 ax + 3y -cz-3 has the solution x = 1, y = - 1 and 2 = 2? о 0 30 o 001 0 - 200 Боо 이 Solve for x 18. 1 0-3 2 x -6 |1 3х-5|
The number of options for a, b, and c that fulfill the system and produce the stated solution is unlimited.
To determine the coefficients a, b, and c such that the system of equations satisfies the given solution x = 1, y = -1, and z = 2, we can substitute these values into the equations and solve for a, b, and c.
Substituting x = 1, y = -1, and z = 2 into the first equation:
a(1) + b(-1) - 3(2) = -1
a - b - 6 = -1
Substituting x = 1, y = -1, and z = 2 into the second equation:
a(1) + 3(-1) - c(2) - 3 = 0
a - 3 - 2c - 3 = 0
a - 2c = 6
Now we have a system of two equations with two unknowns:
a - b - 6 = -1
a - 2c = 6
We can solve this system using standard techniques such as substitution or elimination.
From the first equation, we have a = b - 5. Substituting this into the second equation, we get:
(b - 5) - 2c = 6
b - 2c = 11
So we have the system:
a = b - 5
b - 2c = 11
The values of a, b, and c can be chosen arbitrarily as long as they satisfy these equations. For example, we can choose a = 0, b = 6, and c = -2, which satisfies the system of equations. However, there are infinitely many possible choices for a, b, and c that would yield the given solution.
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URGENT!! PLEASE help and don’t link files!!!
Find Surface Area of triangular shown.
(If you can please explain, thank you!!)
Answer:336cm cube
Step-by-step explanation:
12x4x7 i think im justlearning this to
HELP ME PLEASEEEEEEEEEEEE
Answer:
Step-by-step explanation:
There is really only one solution which is the third choice. When you are dealing with the equation in the form of y = mx + b, the b stands for the y-intercept. So, your y-intercept equals -1. When you are finding the x-intercept you can just substitute y for 0(the definition of x-intercept is where the graph crosses the x-axis, meaning y needs to be equal to 0) and you get that x equals 1/5.
Plzzzzzzzzzz help someone I’m having trouble
Answer: X=8 Y=138
Step-by-step explanation:
Answer: x=2 and y=12
Step-by-step explanation:
Damien LOVES fruit. He has 22 oranges. If Damien, only eats .2 of a cupcake each day, how many days will it take him to eat all 22 cupcakes?
Answer:
c
Step-by-step explanation:
Rogue River Kayaks specializes in making lightweight yet durable kayaks for white water rafting. It is essential that they make sure that their kayaks maintain a certain level of buoyancy even for the largest of kayakers. According to the latest data set provided by National Transportation and Safety Board (NTSB), the weights of men in the United States are normally distributed with a mean of 188.6 pounds and a standard deviation of 38.9 pounds.
Using excel solve the following:
1. Rogue River Kayaks claims that they can fit 99% of men in their Boulder Buster Kayak. What is the upper weight limit the kayak needs to support for the company to meet their claim? Write or type out the formula you used to calculate your answer.
2. The company claims that the ideal weight range for kayakers in the Boulder Buster is between 135 pounds and 210 pounds. What probability a randomly selected rider falls into this range? Use the characteristics from latest data set as mentioned in the paragraph above to answer this question. Write or type out the formula you used to calculate your answer.
3. Why is it important for Rogue River Kayaks to consistently look to the NTSB to update the distribution of weights in the United States?
1. the upper weight limit the kayak needs to support for the company to meet their claim is 270 pounds (rounded off to nearest pound).
2. the probability that a randomly selected rider falls into the weight range of 135-210 pounds is 0.6245 or approximately 62.45%.
3. It is important for Rogue River Kayaks to consistently look to the National Transportation and Safety Board (NTSB) to update the distribution of weights in the United States because the company needs to ensure that their kayaks can accommodate the maximum number of people possible.
As the distribution of weights in the population changes, the company needs to adjust the design of their kayaks to meet the needs of their customers.
By staying up-to-date with the latest data provided by NTSB, Rogue River Kayaks can ensure that their products remain competitive and continue to meet the needs of their customers.
1. We know that the weights of men in the United States are normally distributed with a mean (μ) of 188.6 pounds and a standard deviation (σ) of 38.9 pounds.
Using Excel, we can use the formula NORM.
INV to find the upper weight limit for the kayak to support if Rogue River Kayaks claims that they can fit 99% of men in their Boulder Buster Kayak.
The formula for NORM.
INV is =NORM.INV(probability,mean,standard deviation)
Where probability = 0.99, mean = 188.6, standard deviation = 38.9.
Thus, =NORM.INV(0.99,188.6,38.9)
= 269.60
Therefore, the upper weight limit the kayak needs to support for the company to meet their claim is 270 pounds (rounded off to the nearest pound).
2. To find the probability that a randomly selected rider falls into the ideal weight range for kayakers in the Boulder Buster, we need to find the z-scores for the given weight range and then use the standard normal distribution table.
The z-score formula is:
z = (x - μ) / σ
where x is the weight, μ is the mean and σ is the standard deviation.
For the lower weight limit of 135 pounds, the z-score is z = (135 - 188.6) / 38.9 = -1.382
For the upper weight limit of 210 pounds, the z-score is z = (210 - 188.6) / 38.9 = 0.551
Using the standard normal distribution table, we can find the probability that a z-score falls between -1.382 and 0.551.
The probability is
P(z < 0.551) - P(z < -1.382)
= 0.7088 - 0.0843
= 0.6245
Therefore, the probability that a randomly selected rider falls into the weight range of 135-210 pounds is 0.6245 or approximately 62.45%.
3. It is important for Rogue River Kayaks to consistently look to the National Transportation and Safety Board (NTSB) to update the distribution of weights in the United States because the company needs to ensure that their kayaks can accommodate the maximum number of people possible.
As the distribution of weights in the population changes, the company needs to adjust the design of their kayaks to meet the needs of their customers.
By staying up-to-date with the latest data provided by NTSB, Rogue River Kayaks can ensure that their products remain competitive and continue to meet the needs of their customers.
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Prove that 3^17−3^15+3^13 is divisible by 73.
HELPPPPPP PLEASEEE
Answer:
Well 3^17-3^15+3^13 equals 116,385,579
Now divide that by 73
So 116,385,579 ÷ 73
That equals 1,594,323
Hope this helps
Answer: 3^13*73
Step-by-step explanation:
3^17-3^15+3^13 So you factor out 3^13
=3^13(3^4-3^2+1)
3^4-3^2+1=73
3^13*73
Assume that the random variable X has the first, second, third and fourth moments given as 1, 2, 3, and 4 respectively and let Y = a + bX+cX². Find the correlation coefficient p(X, Y).
The correlation coefficient between X and Y for the given moments is equal to (b + c) / √(b²(1 + 2c) + 2bc²).
Y = a + bX+cX²
To find the correlation coefficient between two random variables X and Y,
Calculate their covariance and standard deviations.
Find the covariance between X and Y.
The covariance between X and Y is ,
cov(X, Y) = E[(X - E[X])(Y - E[Y])]
To calculate this, find the expected values E[X] and E[Y].
Since we are given the first four moments of X,
Use them to find the mean (E[X]) and the variance (Var[X]) of X,
E[X]
= μ
= 1
Var[X]
= E[X²] - (E[X])²
= 2 - 1²
= 2 - 1
= 1
Now let us find E[Y],
E[Y] = E[a + bX + cX²]
= a + bE[X] + cE[X²]
To calculate E[X²], use the second moment of X,
E[X²] = 2
Substituting these values, we have,
E[Y] = a + b(1) + c(2)
Now calculate the covariance,
cov(X, Y)
= E[(X - E[X])(Y - E[Y])]
= E[X·Y - X·E[Y] - E[X]·Y + E[X]·E[Y]]
= E[X·Y] - E[X]·E[Y] - E[X]·E[Y] + E[X]·E[Y]
= E[X·Y] - E[X]·E[Y]
The second moment of XY,
E[XY]
= E[(a + bX + cX²)X]
= E[aX + bX² + cX³]
= aE[X] + bE[X²] + cE[X³]
To calculate E[X³], use the third moment of X,
E[X³] = 3
Substituting these values, we have,
E[XY]
= aE[X] + bE[X²] + cE[X³]
= a(1) + b(2) + c(3)
= a + 2b + 3c
Finally, substitute the expressions for E[XY] and E[X]·E[Y] back into the covariance formula to obtain,
cov(X, Y)
= E[XY] - E[X]·E[Y]
= (a + 2b + 3c) - (1)(a + b(1) + c(2))
= a + 2b + 3c - a - b - 2c
= b + c
Next, calculate the standard deviations of X and Y.
The standard deviation of X is the square root of the variance,
σ(X)
= √Var[X]
= √1
= 1
The standard deviation of Y can be calculated as follows,
Var[Y]
= Var[a + bX + cX²]
= Var[bX + cX²]
= b²Var[X] + c²Var[X²] + 2bcCov[X, X²]
Var[X] and Var[X²] from the given moments,
Var[X] = 1
Var[X²]
= E[X⁴] - (E[X²])²
= 4 - 2²
= 4 - 4
= 0
Substituting these values, we have,
Var[Y]
= b²Var[X] + c²Var[X²] + 2bcCov[X, X²]
= b²(1) + c²(0) + 2bcCov[X, X²]
= b² + 2bcCov[X, X²]
Since Cov[X, X²] = b + c, substitute this back into the equation,
Var[Y]
= b² + 2bc(b + c)
= b² + 2b²c + 2bc²
= b²(1 + 2c) + 2bc²
The standard deviation of Y is the square root of the variance,
σ(Y)
= √Var[Y]
= √(b²(1 + 2c) + 2bc²)
Finally, calculate the correlation coefficient,
p(X, Y)
= cov(X, Y) / (σ(X) · σ(Y))
= (b + c) / (1 · √(b²(1 + 2c) + 2bc²))
= (b + c) / √(b²(1 + 2c) + 2bc²)
Therefore, the correlation coefficient between X and Y is given by (b + c) / √(b²(1 + 2c) + 2bc²).
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A) Set up an integral for finding the Laplace transform of f(t) = 5.
B) Find the antiderivative (with constant term 0) corresponding to the previous part.
C) Evaluate appropriate limits to compute the Laplace transform of f(t).
D) Where does the Laplace transform you found exist? In other words, what is the domain of f(s)?
a. In this case, since f(t) = 5, L{5} = ∫[0 to ∞] 5 * e^(-st) dt. b. the antiderivative simplifies to ∫(5 * e^(-st)) dt = (5/s) * e^(-st). c. the Laplace transform simplifies to (5/s) * (0 - 1).
A) To set up an integral for finding the Laplace transform of f(t) = 5, we can use the definition of the Laplace transform. The Laplace transform of a function f(t) is given by the integral:
L{f(t)} = ∫[0 to ∞] f(t) * e^(-st) dt
where s is the complex frequency parameter. In this case, since f(t) = 5, we have:
L{5} = ∫[0 to ∞] 5 * e^(-st) dt
B) To find the antiderivative corresponding to the previous part, we can integrate the function 5 * e^(-st) with respect to t. The antiderivative, or indefinite integral, of 5 * e^(-st) dt is:
∫(5 * e^(-st)) dt = (5/s) * e^(-st) + C
where C is the constant of integration. Since we are given that the constant term is 0, the antiderivative simplifies to:
∫(5 * e^(-st)) dt = (5/s) * e^(-st)
C) To evaluate the Laplace transform of f(t) = 5, we need to compute the integral from 0 to ∞. Plugging in the antiderivative from part B, we have:
L{f(t)} = ∫[0 to ∞] 5 * e^(-st) dt = lim[T→∞] [(5/s) * e^(-sT) - (5/s) * e^(-s(0))]
As T approaches infinity, the term e^(-sT) goes to 0, since the exponential function decays as the exponent becomes more negative. Therefore, the Laplace transform simplifies to:
L{5} = lim[T→∞] [(5/s) * e^(-sT) - (5/s) * e^(0)]
= (5/s) * (0 - 1)
Simplifying further, we find:
L{5} = -5/s
D) The Laplace transform L{f(t)} = -5/s exists for values of s where the integral converges. The Laplace transform is defined for a certain range of complex numbers, which forms the domain of the Laplace transform. In this case, the Laplace transform of f(t) = 5 exists for all complex numbers s except for s = 0. Therefore, the domain of f(s) is the set of all complex numbers except for s = 0.
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