The volume of the solid that lies within the sphere x² + y² + z² = 36, above the xy-plane, and below the cone z = √(x² + y²) is 9π times the density ρ.
To find the volume of the solid that lies within the sphere x² + y² + z² = 36, above the xy-plane, and below the cone z = √(x² + y²), we need to set up a triple integral in cylindrical coordinates.
Cylindrical coordinates are particularly suitable for this problem because of the symmetry of the sphere and the cone.
In cylindrical coordinates, we have:
x = r cos θ
y = r sin θ
z = z
The sphere equation in cylindrical coordinates becomes:
r² + z² = 36
The cone equation remains the same:
z = √(r²)
To find the limits of integration, we need to determine the region of intersection between the sphere and the cone.
From the cone equation, we have:
z = √(r²) = r
Substituting this into the sphere equation, we get:
r² + r² = 36
2r² = 36
r² = 18
r = √18 = 3√2
So, the limits for r are 0 to 3√2, and for θ, we take a full revolution, 0 to 2π. For z, we take the range from 0 to the cone z = √(r²).
The volume V can be calculated using the triple integral:
V = ∫∫∫ ρ dz dr dθ
Integrating ρ (the density function) over the given limits, we get:
V = ∫[0 to 2π] ∫[0 to 3√2] ∫[0 to √(r²)] ρ dz dr dθ
To evaluate this integral, we consider ρ as a constant factor, as it does not depend on the variables of integration:
V = ρ ∫[0 to 2π] ∫[0 to 3√2] ∫[0 to √(r²)] dz dr dθ
The innermost integral with respect to z evaluates to z evaluated at the limits:
V = ρ ∫[0 to 2π] ∫[0 to 3√2] [√(r²) - 0] dr dθ
Simplifying further:
V = ρ ∫[0 to 2π] ∫[0 to 3√2] r dr dθ
Now, we integrate with respect to r:
V = ρ ∫[0 to 2π] [(r² / 2)] evaluated from 0 to 3√2 dθ
V = ρ ∫[0 to 2π] [(9/2) - 0] dθ
V = ρ ∫[0 to 2π] (9/2) dθ
V = ρ * (9/2) * (θ evaluated from 0 to 2π)
V = ρ * (9/2) * (2π - 0)
V = ρ * (9π)
Therefore, the volume of the solid that lies within the sphere x² + y² + z² = 36, above the xy-plane, and below the cone z = √(x² + y²) is 9π times the density ρ.
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A certain county is 25% African American. Suppose a researcher is looking at jury pools, each with 200 members, in this county. The null hypothesis is that the probability of an African American being selected into the jury pool is 25%. a. How many African Americans would the researcher expect on a jury pool of 200 people if the null hypothesis is true? b. Suppose pool A contains 17 African American people out of 200, and pool B contains 39 African American people out of 200. Which will have a smaller p-value and why?
Answer: a. 50 African Americans
b. Pool B will have a smaller p-value because that pool's number of AA people is further from the hypothesized number of AA people.
Step-by-step explanation:
plz help, i give brainliest
Answer:
A
Step-by-step explanation:
Help with my mathhh!
Answer:
Step-by-step explanation:
d
Answer:
<BOF
Step-by-step explanation:
sum of both these angles is 180 so C is the answer.
How much Interest(in dollars) is earned by Investing $2200 at a simple interest rate of 8% for 12 years? Write the correct answer.
A = $4,312.00
I = A - P = $2,112.00
hey!
Equation:
A = P(1 + rt)
Calculation:
First, converting R percent to r a decimal
r = R/100 = 8%/100 = 0.08 per year.
Solving our equation:
A = 2200(1 + (0.08 × 12)) = 4312
A = $4,312.00
The total amount accrued, principal plus interest, from simple interest on a principal of $2,200.00 at a rate of 8% per year for 12 years is $4,312.00.
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have a good day! hope i helped in some way
For the differential equation s" + bs' +9s = 0, find the values of b that make the general solution overdamped, underdamped, or critically damped. (For each, give an interval or intervals for b for which the equation is as indicated. Thus if the the equation is overdamped for all b in the range -1
The general solution to the differential equation s" + b s' + 9s = 0 can be written as:
[tex]s(t) = c1*e^(-bt/2)*cos(({4b-36)/2)t} 4b-36)/2)t) + c2e^(-bt/2)*sin\sqrt{(4b-36)/2)*t)} (4b-36)/2)*t)[/tex]
where c1 and c2 are constants determined by the initial conditions.
The behavior of the solutions to this equation depends on the value of the parameter b. Specifically, there are three cases to consider:
Overdamped: If b > 6, then the roots of the characteristic equation[tex]s^2 + bs + 9 = 0[/tex] are real and distinct, i.e., [tex]b^2 - 4ac[/tex] > 0. In this case, the general solution is a linear combination of two decaying exponentials, and the system is said to be overdamped. To find the interval for b for which the equation is overdamped, we solve the inequality b > 6, which gives the interval (6, infinity).
Critically damped: If b = 6, then the roots of the characteristic equation are real and equal, i.e., [tex]b^2 - 4ac[/tex]= 0. In this case, the general solution is a linear combination of two decaying exponentials, where one of the exponentials has an additional factor of t. The system is said to be critically damped. To find the interval for b for which the equation is critically damped, we solve the equation b = 6, which gives the singleton set {6}.
Underdamped: If b < 6, then the roots of the characteristic equation are complex conjugates, i.e., [tex]b^2 - 4ac[/tex] < 0. In this case, the general solution is a linear combination of two decaying exponentials, where the exponentials have a sinusoidal factor. The system is said to be underdamped. To find the interval for b for which the equation is underdamped, we solve the inequality b < 6, which gives the interval (-infinity, 6).
Therefore, the interval for b that makes the general solution overdamped is (6, infinity), the singleton set {6} makes the general solution critically damped, and the interval for b that makes the general solution underdamped is (-infinity, 6).
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What is the percent of 0.875?
Answer:
87.5%
Step-by-step explanation:
Just move the decimal place twice to the right.
Answer:
87.5%
Step-by-step explanation:
0.875× 100= 87.5%
PLS HELP PLS HELP PLS HELP PLS HELP PLS HELP I BEG U PLS BOTH QUESTIONS PLS
Answer:14 is obtuse
15 is A
Step-by-step explanation:
For 14 the only remaining angle is over 90
For 15 the first triangle has all same side and angle measures
second triangle has 2 equal side measures
and third has none
Answer:
1: Obtuse
2: A
Step-by-step explanation:
One angle is 57° and another is 12°. If we add them, we get 69. Subtract 69 from 180 to get missing angle: 180-69=111
the missing angle is 111°. We know this is not a right triangle because those have a 90° angle. Since one angle is bigger than 90 (111°), the triangle is obtuse.
For the second one, we see that the first triangle is equal on all sides, so it is equilateral. The second triangle has just two equal sides, so it is isosceles. The thrid one is different side length on all of them, so it is scalene. the answer is A.
Honestly help I’m slow
Answer:
46 degrees
Step-by-step explanation:
x equals 23 so just multiply by 2
Answer:
[tex]46^{o}[/tex]
Step-by-step explanation:
Bottom-right angle is 65 as 180 - 90 - 25 = 65
Top-left angle is also 65 as angles opposite a point are equal (when divided by straight lines)
Angles around a point = 360 so 360 - 65 - 65 - 90 - 25 = 115
5x = 115
x = 23
23 x 2 = 46
Is the President doing a good job? We will examine this by taking a random sample of n = 4220 adults and asking whether they feel the president is doing a good job? Of these sampled adults, x = 2222 said the President was doing a good job. (Assume nobody lies.) Let p be the (unknown) true proportion of adults who feel the President is doing a good job. We want to estimate p. X is the random variable representing the number of sampled adults who say the president is doing a good job. have?
a) What type of probability distribution does X have?
O binomial
O gamma exponential
O Weibull
O Poisson
b) What was the sample proportion, P^, of sampled adults who say the President is doing a good job? _____
b) What is the R formula for the expected value of X in terms of n and p?
O sqrt(n*p*(1-p))
O n*p*(1 - p)
On^2
O n*p
O 1/p
d) What is the z critical value that we would use to construct a classical 90% confidence interval for p? _______
e) Construct a 90% classical confidence interval for p? (_____,_____)
f) How long is the 90% classical confidence interval for p? ______
g) If we are creating a 90% classical confidence interval for p based upon the sample size of 4220, then what is the longest possible length of this interval? _____
(a) Binomial probability distribution does X have. The option 1 is correct answer.
(b) 0.5265 is the sample proportion [tex]\hat{P}[/tex] of sampled adults who say the President is doing a good job.
(c) n * p is the R formula for the expected value of X in terms of n and p. The option 4 is correct answer.
(d) 1.645 is the z critical value that we would use to construct a classical 90% confidence interval for p.
(e) A 90% classical confidence interval for p is 0.5104, 0.5428.
(f) 0.0324 is the 90% classical confidence interval for p.
(g) 0.0357 is the longest possible length of this interval.
a) The random variable X, representing the number of sampled adults who say the President is doing a good job, follows a binomial probability distribution. Therefore, the correct answer is option 1.
b) The sample proportion, [tex]\hat{P}[/tex], of sampled adults who say the President is doing a good job can be calculated by dividing the number of adults who said the President was doing a good job (x = 2222) by the total sample size (n = 4220):
[tex]\hat{P}[/tex] = x / n
= 2222 / 4220
= 0.5265
c) The expected value of X is given by
n*p,
where n is the sample size and
p is the true proportion of adults who feel the President is doing a good job.
Therefore, the correct answer is option 4.
d) To construct a classical 90% confidence interval for p, we need to find the z critical value. This value can be found using a z-table or calculator and is approximately 1.645.
e) Using the sample proportion, [tex]\hat{P}[/tex], the z critical value, and the sample size, a 90% classical confidence interval for p can be calculated. This is done using the formula:
[tex]\hat{P} \pm z \times \sqrt{\frac{\hat{P} \times (1 - \hat{P})}{n}}[/tex]
The interval is (0.5104, 0.5428).
f) The length of the 90% classical confidence interval for p can be found by subtracting the lower limit from the upper limit: 0.5428 - 0.5104 = 0.0324.
g) The longest possible length of the 90% classical confidence interval for p can be found by using the formula:
[tex]2z \sqrt{\frac{\hat{P} ( 1 - \hat{P})}{n}[/tex]
Plugging in the values from the sample, we get
21.645 √(0.5266(1-0.5266)/4220)
= 0.0357.
This means that the interval can be at most 0.0357 in length.
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Nikolai was curious if he could predict how many "likes" an internet video received based on how many views the video had. He took a random sample of videos and noticed a positive linear relationship between number of views and number of likes (both in thousands). Here is computer output from a least-squares regression analysis on his data:
Which of these is an appropriate least-squares equation for this model?
Answer:
likes
=0.420+0.034(views) (B) on khan academy
Step-by-step explanation:
the expression 8+2t can be used to find the total cost of admission and t amusement rides at a country fair. what statement is true?
the anwser is going to be A
SURFACE AREA!!
can someone please help me get the answer on these two I’m stuck
Answer:
96 i think :)
Step-by-step explanation:
15x4=60
6x6=36
60+36=96
HELP ME ASAP PLEASE!!!!! LOOK AT SCREENSHOT (10 PTS)
Answer: i would say A and D but i'm not sure of the others
what is the sum of 3x4
Answer:
12
Step-by-step explanation:
3 4's
3+3+3+3=12
Hope that helps :)
Carissa’s gerbil has a tail that is the same length as its body length. Its tail is 102 millimeters. How long is her gerbil in centimeters?
Answer:
...
Step-by-step explanation:
the answer is 10.2 :)
How many significant figures will there be in the answer to the following problem? You do not have to solve the problem. 3.4 • 17.05 =
Answer:
3.4 × 17.015 = 58
Step-by-step explanation:
3.4 → two non-zero digits = two sig figs
17.05 → four non-zero digits = four sig figs
- hope this helps!
A bag contains x counters. 7 of the counters are blue. Sam takes at random a counter from the bag and does not replace it. Jill then takes a counter from the bag. The probability they both take a blue counter is 0.2. Form an equation involving x, in the form x + bx +c=0.
Answer:
0.2x² - 0.2x - 42 = 0
Step-by-step explanation:
Number of counters = x
Number of blue counters =7
Probability, p = required outcome / Total possible outcomes
P(Sam picks blue ) : = 7/x
P(jill picks blue) = 6/(x - 1)
P(Sam picks blue) * P(Jill picks blue) = 0.2
7/x * 6/(x-1) = 0.2
42/x(x - 1) = 0.2
42 / x² - x= 0.2
42 = 0.2(x² - x)
42 = 0.2x² - 0.2x
0.2x² - 0.2x - 42
Please help I need help and please explain
Answer:
click c
Step-by-step explanation:
and its right
A square has a perimeter of 36 inches and a smaller square has a side length of 4 inches. What is the ratio of the areas of the larger square to the smaller square?
Answer:
3:2
Step-by-step explanation:
a square has the same side lengths so just √36 = 6 to find the sides of the squares then compare the two sides in ratio form 6:4 then reduce
Gina and Rhonda work for different real estate agencies. Gina earns a monthly salary of $5,000 plus a 6% commission on her sales. Rhonda earns a monthly salary of $6,500 plus a 4% commission on her sales. How much must each sell to earn the same amount in a month?
Answer: Gina and Rhonda work for different real estate agencies. Gina earns a monthly salary of $5,000 plus a 6% commission on her sales. Rhonda earns a monthly salary of $6,500 plus a 4% commission on her sales. How much must each sell to earn the same amount in a month?
Step-by-step explanation:
$5,000 + 6% + $6,500 + 4% = 11500.1 or 11500
There were 32 volunteers to donate blood. Unfortunately, n of the volunteers did not meet the health
requirements, so they couldn't donate. The rest of the volunteers donated 470 milliliters each.
How many milliliters of blood did the volunteers donate?
Write your answer as an expression.
x(x-4)=12 solve for x
Answer:
x=6 and x=-2
Step-by-step explanation:
so
x(x-4)=12
first distribute
then move the terms
and the u get
x=6 and x=-2
hope this helped
Answer:
x=6, x=-2
Step-by-step explanation:
x(x-4)=12
distributive property, x^2-4x=12
x^2-4x-12=0
(x-6)(x+2)
therefore, x=6, x=-2
Let λ be an eigenvalue of an invertible matrix a. show that λ^−1 is an eigenvalue of A^−1. [hint: suppose a nonzero x satisfies Ax=λx.]
Let λ be an eigen value of an invertible matrix. Then, [tex]\lambda^{-1}[/tex] is surely an eigenvalue of [tex]A^{-1}[/tex].
What is an invertible matrix?
For a matrix to be invertible, it must have a unique matrix that, when multiplied with the original matrix, gives the identity matrix.
[tex]A * B = B * A = I[/tex]
Suppose A is an invertible matrix and λ is an eigenvalue of A with a corresponding nonzero eigenvector x, i.e., Ax = λx.
To show that [tex]\lambda^{-1}[/tex] is an eigenvalue of [tex]A^{-1}[/tex], we need to find a nonzero vector y such that [tex]A^{-1}y[/tex] = [tex]\lambda^{-1}y[/tex].
Let's start by multiplying both sides of the equation Ax = λx by [tex]A^{-1}[/tex]:
[tex]A^{-1}(Ax) = A^{-1}(\lambda x)[/tex]
(x is nonzero, so we can divide by x)
[tex]A^{-1}(Ax/x) = A^{-1}(\lambda x/x)\\A^{-1}(A(x/x)) = A^{-1}(\lambda)[/tex]
Since [tex]A^{(-1)}A = I[/tex] (identity matrix), and x/x = 1, we have:
[tex]A^{(-1}(I) = A^{(-1)}[/tex] λ
[tex]A^{(-1)}[/tex] = λ[tex]A^{(-1)}[/tex]
Now, let y = A^(-1)x. We can rewrite the equation above as:
[tex]A^{(-1)}x[/tex] = λ[tex]A^{(-1)}y[/tex]
([tex]A^{(-1)}x[/tex]/λ) = [tex]A^{(-1)}y[/tex]/λ
(x is nonzero, so we can divide by x)
([tex]A^{(-1)}x/x[/tex])/λ = [tex](A^{(-1)}y/y[/tex])/λ
([tex]A^{(-1)}(x/x)[/tex])/λ = ([tex]A^{(-1)}y/y[/tex])/λ
([tex]A^{(-1)}(1)[/tex])/λ = ([tex]A^{(-1)}y/y[/tex])/λ
([tex]A^{(-1)}[/tex])/λ = ([tex]A^{(-1)}y[/tex])/λ
Since [tex]A^{(-1)}[/tex] is a matrix and λ is a scalar, we can rearrange the equation as follows:
([tex]A^{(-1)}[/tex])/λ = [tex]A^{(-1)}[/tex]y/λ
(1/λ)[tex]A^{(-1)}[/tex] =[tex]A^{(-1)}[/tex]y/λ
This shows that 1/λ is an eigenvalue of [tex]A^{(-1)}[/tex] with the corresponding eigenvector y. Therefore, we have shown that if λ is an eigenvalue of A, then [tex]\lambda^{(-1)}[/tex] is an eigenvalue of [tex]A^{(-1)}[/tex]
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Do the ordered pairs below represent a relation, a function, both a relation and a function, or neither a relation nor a function?
(-5,9) , (0,-1) , (7,-15) , (9,-19)
Option D is correct.
The relation {(-8, -6) (-5, 2) (-8, 1) (7, 3)} is not a function.
Step-by-step explanation:
Given the relation: {(-8, -6) (-5, 2) (-8, 1) (7, 3)}
Domain is the set of all possible inputs of a relation i.e { -8, -5 , -8 , 7}
Range is the set of output values of a function i.e, {-6, 2 , 1 , 3}
The mapping as shown below in the figure:
A function is a relation in which every element of the domain is matched to not more than one element of the range.
In other words, we can say that ,no value of x gets mapped to more than 1 value of y.
Since, from the mapping you can see that the domain value -8 paired with -6 and 1; as x is used more than once.
Therefore, this relation is not a function
What is the value of 9x^2 + 13x – 15, if x = 2
Step-by-step explanation:
Given,
[tex]9 {x}^{2} + 13x - 15[/tex]
and
[tex]x = 2[/tex]
Substitute x = 2 into expression.
[tex]9 {x}^{2} + 13x - 15 = 9( {2}^{2} ) + 13(2) - 15 \\ = 9(4) + 26 - 15 \\ = 36 + 26 - 15 \\ = 62 - 15 \\ = 47[/tex]
Answer:
9(2)×2+13(2)-15
18×2+26-15
36+11
47 is your answer ☺️☺️☺️
help please! need an answer asap................
Answer:
x ≈ 121.2 ft
Step-by-step explanation:
Using the tangent ratio in the right triangle
tan60° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{x}{70}[/tex] ( multiply both sides by 70 )
70 × tan60° = x , then
x ≈ 121.2 ft ( to 1 dec. place )
"
Use the non- linear shooting method with accuracy 10^-1 (stop at 2nd iteration if this accuracy is not attained earlier) to solve the boundary-value problem: y"=-yy'+y, and 1<=x<=2, y(1)=1/2,y(2) =1/3. use h =0.5.Compare your results with actual solution : y(x) =1/ (x+1).
The non-linear shooting method with an accuracy of [tex]10^{-1}[/tex] was applied to solve the boundary-value problem y" = -yy' + y. The results were compared with the actual solution y(x) = 1/(x+1).
To solve the given boundary-value problem using the shooting method, we consider the problem as an initial-value problem by introducing an initial condition for y'(1).
Then, an iterative process is performed to find the appropriate value of y'(1) that satisfies the second boundary condition at x = 2.
Starting with an initial guess for y'(1), say y'(1) = a, we integrate the differential equation y" = -yy' + y numerically over the interval 1 <= x <= 2 using a step size of h = 0.5.
The numerical integration can be done using methods such as the Runge-Kutta method.
At each iteration, we compare the computed value of y(2) with the desired boundary condition y(2) = 1/3. If the accuracy of [tex]10^{-1}[/tex] is not attained after the second iteration, the process is stopped.
If the accuracy is achieved, the solution is considered as the actual solution.By comparing the obtained numerical solution with the actual solution y(x) = 1/(x+1), we can evaluate the accuracy of the non-linear shooting method.
The difference between the two solutions can be analyzed to assess the effectiveness of the method in solving the given boundary-value problem.
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25x+20y=200 in slope intercept form.
Answer:
5x+4y-40=0
Step-by-step explanation:
HELP PLEASE‼️
There are 148 legs in a farm yard full of goats and chickens. There are 62 total animals. How many of each are there.
It is 12 goats and 50 chickens to make those animals and that exact number of legs
12. from the slope of your best-fit line, what is the velocity of the pacific plate, as expressed in cm/yr? (2 significant figures required)
The velocity of the Pacific plate, expressed in centimeters per year (cm/yr), can be determined from the slope of the best-fit line in a geologic study.
In a geologic study, if data points representing the position of the Pacific plate are collected over a period of time, a best-fit line can be calculated to represent the trend of plate movement.
- The slope of this line represents the rate of change of position over time, which corresponds to the velocity of the plate. By examining the slope of the best-fit line and converting it to centimeters per year, we can determine the velocity at which the Pacific plate is moving.
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