Step 1: The intersection curve is only at the origin.
Step 2: Set up the volume integral over the sphere region.
Step 3:Evaluate the integral to find the volume: [tex]\(V = 216\pi\sqrt{2}\).[/tex]
To find the volume of the solid enclosed by the cone [tex]\(z = x^2 + y^2\)[/tex] and the sphere[tex]\(x^2 + y^2 + z^2 = 72\),[/tex] we'll first find the intersection curve of the cone and the sphere in cylindrical coordinates, then set up the triple integral to find the volume.
Step 1: Finding the intersection curve:
The cone equation in cylindrical coordinates becomes [tex]\(z = r^2\)[/tex] and the sphere equation remains the same as [tex]\(r^2 + z^2 = 72\).[/tex]
To find the intersection, we set these two equations equal to each other:
[tex]\[r^2 = r^2 + z^2\][/tex]
Substitute [tex]\(z = r^2\)[/tex] from the cone equation:
[tex]\[r^2 = r^2 + (r^2)^2\][/tex]
[tex]\[r^2 = r^2 + r^4\][/tex]
[tex]\[r^4 = 0\][/tex]
From this, we see that the only solution is (r = 0), which corresponds to the point at the origin.
Step 2: Setting up the integral for volume:
We'll integrate over the region where the cone lies within the sphere, which is the entire sphere. In cylindrical coordinates, the limits of integration are [tex]\(0 \leq r \leq 6\)[/tex] (since the sphere has radius [tex]\(\sqrt{72} = 6\)) and \(0 \leq \theta \leq 2\pi\).[/tex]
The limits for \(z\) are from the cone to the sphere, so it's from [tex]\(r^2\) to \(\sqrt{72-r^2}\).[/tex]
Thus, the volume integral is:
[tex]\[V = \iiint_{E} r \, dz \, dr \, d\theta\][/tex]
Where (E) is the region enclosed by the cone and the sphere.
Step 3: Evaluate the integral:
[tex]\[V = \int_{0}^{2\pi} \int_{0}^{6} \int_{r^2}^{\sqrt{72-r^2}} r \, dz \, dr \, d\theta\][/tex]
Let's evaluate this integral step by step:
[tex]\[V = \int_{0}^{2\pi} \int_{0}^{6} (r\sqrt{72-r^2} - r^3) \, dr \, d\theta\][/tex]
[tex]\[V = \int_{0}^{2\pi} \left[-\frac{1}{4}(72-r^2)^{3/2} - \frac{1}{4}r^4\right]_{0}^{6} \, d\theta\][/tex]
[tex]\[V = \int_{0}^{2\pi} \left[-\frac{1}{4}(0)^{3/2} - \frac{1}{4}(6^4) - \left(-\frac{1}{4}(72)^{3/2} - \frac{1}{4}(0)^4\right)\right] \, d\theta\][/tex]
[tex]\[V = \int_{0}^{2\pi} \left[\frac{1}{4}(72)^{3/2}\right] \, d\theta\][/tex]
[tex]\[V = \frac{1}{4}(72)^{3/2} \int_{0}^{2\pi} 1 \, d\theta\][/tex]
[tex]\[V = \frac{1}{4}(72)^{3/2} \cdot 2\pi\][/tex]
[tex]\[V = 36\pi \sqrt{72}\][/tex]
[tex]\[V = 36\pi \times 6\sqrt{2}\][/tex]
[tex]\[V = 216\pi\sqrt{2}\][/tex]
So, the volume of the solid enclosed by the cone and the sphere is [tex]\(216\pi\sqrt{2}\).[/tex]
105,159 rounded to the nearest ten thousand
7/2x-2=28-4x solve for x
The total interest paid on a 3-year loan at 9% interest compounded monthly is $1505.82 determine the monthly payment for the loan.
Steve can complete the 100m dash in 10 seconds while Paul can run it in 12 seconds. How does Steve's time compare to Paul's?
Given that x has a Poisson distribution with
mu
μ
equals
=
13
13, what is the probability that x
equals
=
5
5?
P(
5
5)
almost equals
≈
0.9930
0.9930 (Round to four decimal places as needed.)
a local charity held a crafts fair selling donated,handmade items. Total proceeds from the sale were $1,875. A total of 95 items were sold,some at $15 each and the rest at $25 each. Let x be the number of $15 items and y the number of $25 items. How many items sold at $25?
Answer:
It’s 30
Step-by-step explanation:
Which figure has the correct lines of symmetry drawn in?
The figure which has the correct lines of symmetry drawn is figure D.
What is line of symmetry?A line of symmetry refers to the line that divides a shape or an object into two equal and symmetrical parts.
The figure given is a five sided shape referred to as a pentagon.
A pentagon has five lines of symmetry.
Therefore, the correct line of symmetry is 5 lines drawn from each angle respectively.
Read more on line of symmetry:
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You have a 20-foot ladder that you are using as you paint your house. You place the ladder so that it forms an angle of 30 at the point of contact between the ladder and the ground. How high will the top of the ladder be above the ground
Answer:
The Ladder is 10 ft high.
Step-by-step explanation
According to the question we have a 20 ft ladder inclined against a wall at a 30° angle, and are asked to find how high the top of the ladder is from the ground. I have added a visual representation to help you better understand the situation. As you can see we need to solve for x. Since we are given one angle and the hypotenuse we can solve for x using the SIN operator.
[tex]sin(a) = \frac{opposite}{hypotenuse}[/tex]
Since we have the angle (a) and the hypotenuse we can just plug the information into the sin equation and solve for the opposite (x).
[tex]sin(30) = \frac{x}{20ft}[/tex]
[tex]sin(30)*20ft = x[/tex]
[tex]0.5*20ft = x[/tex]
[tex]10ft = x[/tex]
So now we can see that the height of the top of the ladder to the ground is 10 ft
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
Suppose a basketball player has made 184 out of 329 free throws. If the player make the next two free throws, I will pay you $24. Otherwise you pay me $12. Find the expected value of the proposition
Final answer:
Expected value of proposition will be 20.134.
Explanation:
Given that the player made 184 out of 329 throws, the probability of making the next throw will be:
P(x)=[Number of shots made]/[Total number of throws]
=184/329
=0.559
Thus the expected value of proposition will be:
0.599 x 24+0.559 x 12
=20.134
Consider a company that selects employees for random drug tests. The company uses a computer to randomly select employee numbers that range from 1 to
6857
6857. Find the probability of selecting a number divisible by 1000. Find the probability of selecting a number that is not divisible by 1000.
The probability of selecting a number divisible by 1000 from the range of 1 to 6857 is approximately 0.0008763. The probability of selecting a number that is not divisible by 1000 is approximately 0.9991237.
Explanation:To find the probability of selecting a number divisible by 1000, we need to determine the total number of possible outcomes and the number of favorable outcomes. There are a total of 6857 possible outcomes since the company selects employee numbers ranging from 1 to 6857. For a number to be divisible by 1000, it must end with three zeros, so we need to find how many numbers satisfy this condition between 1 and 6857.
Since 1000 is the smallest number divisible by 1000, we can calculate the number of favorable outcomes by finding the largest integer that satisfies the condition. In this case, it is 6000. Therefore, there are 6 favorable outcomes.
The probability of selecting a number divisible by 1000 is calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Favorable outcomes / Total outcomes = 6 / 6857 ≈ 0.0008763.
The probability of selecting a number that is not divisible by 1000 can be calculated as the complement of the probability of selecting a number divisible by 1000:
Probability(not divisible by 1000) = 1 - Probability(divisible by 1000) ≈ 1 - 0.0008763 ≈ 0.9991237.
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6x2=(3×2)×___=___
6×4=2×(6×___)=___
2×(6×4)=____×8=____
100 meter long Christmas train needs 30 seconds to cross a 400nmetrr long bridge assuming the train goes at a steady speed how fast is it
EASY 5 POINTS!! Which expression gives the correct volume of the figure?
Answer:
(3×2×1)+(3×1×1)
Step-by-step explanation:
AS you can se ein the figure, the figure is not a solid full rectangula prism, it is made out of two prisms the first one is the 3x2x1 rectangular prism, and the second is the 3x1x1 square prism, so by adding the volume of those two you can figure out the volume of the whole figure that is why the answer to the problem is (3×2×1)+(3×1×1)
James deposits $600 in an account that pays 4% simple interest, and $1200 in a second account which has a higher interest rate but is more risky. What interest rate must he get on the second account in order to earn at least $132 in interest for the year?
Which pairs of triangles are similar? Check all that apply.
Answer:
option (2) and (5) are correct.
ΔABC ≅ ΔJLK
ΔDEF ≅ ΔGHI
Step-by-step explanation:
Given four right angled triangle with measure of sides.
We have to check for the pairs of triangle to be similar.
Two triangles are said to be similar if their corresponding angles are equal or their corresponding sides are same ratio.
Consider, ΔABC and ΔJLK.
∠C = ∠L = 90° (given)
Also ratio of corresponding sides are same ratio, that is
[tex]\frac{AC}{LJ}=\frac{14}{7}=\frac{2}{1}[/tex]
Also, [tex]\frac{CB}{KL}=\frac{20}{10}=\frac{2}{1}[/tex]
Thus, ΔABC ≅ ΔJLK.
Option (5) is correct.
Consider, ΔDEF and ΔGHI.
∠I = ∠F = 90° (given)
Also ratio of corresponding sides are same, that is
[tex]\frac{DF}{GI}=\frac{8}{12}=\frac{2}{3}[/tex]
Also, [tex]\frac{EF}{HI}=\frac{10}{15}=\frac{2}{3}[/tex]
Thus, ΔDEF ≅ ΔGHI.
Option (2) is correct.
Thus, option (2) and (5) are correct.
value for X^2 left for 95% confidence interval when n=18
g(x) = x2 + 2, find g(3).
Answer:
Plug in 3 for x on the right side
3^2+2,
3 squared equals 9 plus 2 equals 11
Final answer: 11
Step-by-step explanation:
.
The coordinates of △ABC△ABC are A(12,8), B(10,18), C(4,16)A(12,8), B(10,18), C(4,16). After a dilation, the coordinates are A'(6,4), B'(5,9), C'(2,8)A′(6,4), B′(5,9), C′(2,8). Find the scale factor.
Answer: [tex]\dfrac{1}{2}[/tex]
Step-by-step explanation:
The dilation is a transformation which makes similar shapes.
We know that In two similar geometric figures, the ratio of their corresponding corresponding x-coordinate is known as the scale factor.
For the given situation, the x-coordinate of A in pre-image = 12
The x-coordinate of A' in image = 6
Then , the scale factor for the dilation is given by :-
[tex]k=\dfrac{6}{12}=\dfrac{1}{2}[/tex]
Hence, the scale factor = [tex]\dfrac{1}{2}[/tex]
A geometry class has a total of 34 students. The number of males is 14 more than the number of females. How many males and how many females are in the class?
A taxi cab charges $0.55 per mile in addition to a $1.75 flat rate fee. Susie has $10 to spend on a taxi cab ride. The taxi driver will not give anyone a ride unless they are going somewhere that is more than 2 miles away. Model Susie’s situation with a system of inequalities.
What is the value of -2 (7-15) over 4
Can the polynomial below be factored into a perfect square? If not, select the answer that best describes why not.
64x^2+49x+8
A.
The x^2 coefficient does not permit the factoring.
B.
The x^2 coefficient does permit the factoring, but the x coefficient does not permit the factoring.
C.
The constant value does not permit the factoring.
D.
The polynomial may be factored into a perfect square.
Answer:
Option B is correct.
Step-by-step explanation:
We will work with the formula : [tex](a+b)^{2}[/tex]
= [tex]a^{2}+2ab+b^{2}[/tex]
Given polynomial is :
[tex]64x^{2} +49x+8[/tex]
here a = [tex]\sqrt{64x^{2} } =8x[/tex]
b = [tex]\sqrt{8}= 2\sqrt{2}[/tex]
2ab = [tex]2*8x*2\sqrt{2} =32\sqrt{2}x[/tex]
Now, we can see that the middle term should be [tex]32\sqrt{2} x[/tex] but in the question, it is given 49x
So, option B is true that - The x^2 coefficient does permit the factoring, but the x coefficient does not permit the factoring.
simplify the expression below. show your work. -(-7y+12)
Write an equation for the function. Tells what each variable you use represents. A plant's height is 1.6 times its age in months.
Each figure is filled with unit cubes. Drag 22 figures whose combined volume is equal to 45 unit cubes.
To solve this problem, what we have to do first is to calculate for the volumes of each figure. Let us start from the figure on the left going to the right. Let us call the figures 1 to 4. Formula for volume is:
V = l * w * h
Figure 1:
V1 = 4 * 3 * 1 = 12 unit cubes
Figure 2:
V2 = 3 * 2 * 3 = 18 unit cubes
Figure 3:
V3 = 2 * 2 * 4 = 16 unit cubes
Figure 4:
V4 = 3 * 3 * 3 = 27 unit cubes
The two combinations that would result in 45 total unit cubes would be:
V = V2 + V4 = 18 unit cubes + 27 unit cubes = 45 unit cubes
So the answers are:
Figure 2 and Figure 4
2w+2l=24 what is the value of l
What is the sum of the arithmetic series? ∑[i=1,15,Fn=5i-1]
The graph of a limacon curve is given. Without using your graphing calculator, determine which equation is correct for the graph.[-5, 5] by [-5, 5] (1 point)
Answer:
r = 2 - 3 cos θ
Step-by-step explanation:
This is the best I could come up with.
Information about movie ticket sales was printed in a movie magazine. in a sample of of fifty pg-rated movies, 36% had ticket sales in excess of $30,000,000. in a sample of thirty-five r-rated movies, 23% grossed over $30,000,000. suppose that this data is used to test the claim that the proportion of movies with ticket sales in excess of $30,000,000 is the same for pg movies as it is for r-rated movies. what would be the test statistic for this test? z = 1.29 z = 2.07 z = 4.005 z = 2.58
Let us say that the samples for fifty movies is 1 and samples for thirty five is 2. To solve the test statistic z, we can use the formula:
z = (p1 – p2) / sqrt [p (1 – p) (1 / n1 + 1 / n2)]
Where,
p1 = is the proportion for sample 1 = 0.36
p2 = is the proportion for sample 2 = 0.23
n1 = number of samples = 50
n2 = number of samples = 35
while p can be calculated using the formula:
p = [p1 * n1 / (n1 + n2)] + [p2 * n2 / (n1 + n2)]
p = [0.36 * 50 / (50 + 35)] + [0.23 * 35 / (50 + 35)]
p = 0.211764705 + 0.094705882
p = 0.306470587
Going back for the calculation of z:
z = (0.36 – 0.23) / sqrt [0.306 (1 – 0.306) (1 / 50 + 1 / 35)]
z = 1.279
Therefore the nearest answer is:
z = 1.29
A teacher has 3 hours to grade all the papers submitted by the 35 students in her class. She gets through the first 5 papers in 30 minutes. How much faster does she have to work to grade the remaining papers in the allotted time?