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Margarita [4]
3 years ago
13

Find a parametric representation for the surface. The part of the sphere x2 + y2 + z2 = 144 that lies between the planes z = â’6

and z = 6. (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of θ and/or Ď•.) (where â’6 < z < 6)

Mathematics
1 answer:
ryzh [129]3 years ago
8 0

We can use the standard set of parametric equations in spherical coordinates:

\begin{cases}x(u,v)=12\cos u\sin v\\y(u,v)=12\sin u\sin v\\z(u,v)=12\cos v\end{cases}

with 0\le u\le2\pi. Then the part of the sphere that lies between the planes z=-6 and z=6 is determined by the domain of v. To find that, we can sketch a picture of the situtation (attached below).

The right triangle in the image has its hypotenuse coinciding with the radius of the sphere so its length is 12, and the vertical leg has length 6 because it coincides with the plane z=6. The angle between them is the upper limit of v. At any point along the intersection of the sphere with the plane z=6, we want to have

\cos v=\dfrac6{12}=\dfrac12\implies v=\dfrac\pi3

and so by symmetry, we need to set -\dfrac\pi3\le v\le\dfrac\pi3

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Answer:

a₁ = 38

Step-by-step explanation:

Given AP, where:

  • 8a₈ = 12a₁₂
  • d = -2,
  • a₁ = ?

=============

<h3>Solution</h3>

aₙ = a₁ + (n-1)d

  • a₈ = a₁ + 7d = a₁ + 7*(-2) = a₁ - 14
  • a₁₂ = a₁ + 11d = a₁ + 11*(-2) = a₁ - 22

8a₈ = 12a₁₂

  • 8(a₁ - 14) = 12(a₁ - 22)
  • 2(a₁ - 14) = 3(a₁ - 22)
  • 2a₁ - 28 = 3a₁ - 66
  • 3a₁ - 2a₁ = -28 + 66
  • a₁ = 38
6 0
3 years ago
Which statement most accurately describes what happens when a bar of chocolate melts? A. A physical change takes place that caus
wariber [46]

Answer:

(B). A physical change takes place that causes molecules to move faster

Step-by-step explanation:

When a bar of chocolate melts its physical form is changed, there is no difference in its chemical composition as it is still same.

When the chocolate bar was frozen its molecules were held strongly, and they were barely moving, whereas when it melts the molecules are more free to move now.

5 0
2 years ago
10x +7 = -3х – 3 help guys
Pepsi [2]

Answer:

x = -10/13

Step-by-step explanation:

10x + 7 = -3x - 3

Add 3x to both sides

10x + 3x + 7 = -3x + 3x - 3

13x + 7 = 0x - 3

13x + 7 = -3

Subtract 7 from both sides

13x + 7 - 7 = -3 - 7

13x + 0 = -10

13x = -10

Divide both sides by 13

13x ÷ 13 = -10 ÷ 13

1x = -10/13

x = -10/13

6 0
3 years ago
Use the Distributive Property to<br> simplify the following:<br> 7(y + 3) =
Ksju [112]

Answer:

7y+21

Step-by-step explanation: i solve it hope this help.

8 0
2 years ago
3x - y = 0<br> 2x - y = 1
9966 [12]
Answer: See below

Explanation:
Solve this system of equation using elimination method:

3x - y = 0
2x - y = 1
________
3x - y = 0
-1(2x - y = 1)
_________
3x - y = 0
-2x + y = -1
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X = -1

We know that x = -1 now let’s plug it in the one of the equation to find y:

2x - y = 1
2(-1) - y = 1
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-y = 3
y = -3

8 0
3 years ago
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