Question 1

Find an equation of the sphere with center $( 7 , - 5,7 )$ that touches the $x y$-plane.

Accepted Answer

A)  $( x + 7 ) ^ { 2 } + ( y - 5 ) ^ { 2 } + ( z + 7 ) ^ { 2 } = 49$ 
B)  $( x + 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$ 
C)  $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 40$ 
D)  $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$ 
E)  $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 7$ 
A)  $( x + 7 ) ^ { 2 } + ( y - 5 ) ^ { 2 } + ( z + 7 ) ^ { 2 } = 49$ 
B)  $( x + 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$ 
C)  $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 40$ 
D)  $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$ 
E)  $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 7$ D

Question 2

A woman walks due west on the deck of a ship at $6 \mathrm { mi } / \mathrm { h }$. The ship is moving north at a speed of $20 \mathrm { mi } / \mathrm { h }$. Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

Accepted Answer

20.9

Question 3

Find an equation of the set of all points equidistant from the points $A ( 7 , - 8 , - 9 )$ and $B ( - 6,4 , - 12 )$.

Accepted Answer

A)  $- 13 x - 12 y + 3 z = - 1$ 
B)  $13 x - 12 y + 3 z = - 1$ 
C)  $- 13 x - 12 y + 3 z = 1$ 
D)  $13 x - 12 y + 3 z = 1$ 
E)  $13 x - 12 y = 0$ 
A)  $- 13 x - 12 y + 3 z = - 1$ 
B)  $13 x - 12 y + 3 z = - 1$ 
C)  $- 13 x - 12 y + 3 z = 1$ 
D)  $13 x - 12 y + 3 z = 1$ 
E)  $13 x - 12 y = 0$ B

Question 4

Let $\mathbf { v } = 7 \mathbf { j }$ and let $\mathbf { u }$ be a vector with length 5 that starts at the origin and rotates in the $x y$ - plane. Find the maximum value of the length of the vector $| \mathbf { u } \times \mathbf { v } |$.

Accepted Answer

A)  $| \mathbf { u } \times \mathbf { v } | = 5$ 
B)  $| \mathbf { u } \times \mathbf { v } | = 7$ 
C)  $| \mathbf { u } \times \mathbf { v } | = 35$ 
D)  $| \mathbf { u } \times \mathbf { v } | = 157$ 
E)  $| \mathbf { u } \times \mathbf { v } | = 1$ 
A)  $| \mathbf { u } \times \mathbf { v } | = 5$ 
B)  $| \mathbf { u } \times \mathbf { v } | = 7$ 
C)  $| \mathbf { u } \times \mathbf { v } | = 35$ 
D)  $| \mathbf { u } \times \mathbf { v } | = 157$ 
E)  $| \mathbf { u } \times \mathbf { v } | = 1$

Question 5

Find the standard equation of the sphere with center C and radius r. $C ( 4 , - 1,5 ) ; r = 8$

Accepted Answer

A)  $( x + 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 5 ) ^ { 2 } = 64$ 
B)  $( x + 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 5 ) ^ { 2 } = 8$ 
C)  $( x - 4 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 5 ) ^ { 2 } = 8$ 
D)  $( x - 4 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 5 ) ^ { 2 } = 64$ 
A)  $( x + 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 5 ) ^ { 2 } = 64$ 
B)  $( x + 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z + 5 ) ^ { 2 } = 8$ 
C)  $( x - 4 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 5 ) ^ { 2 } = 8$ 
D)  $( x - 4 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 5 ) ^ { 2 } = 64$

Question 6

Find the midpoint of the line segment joining the given points. $( 4 , - 2 , - 3 )$ and $( 6 , - 4 , - 1 )$

Accepted Answer

A)  $( 5 , - 3 , - 2 )$ 
B)  $( - 1 , - 3 , - 1 )$ 
C)  $( - 1,1 , - 1 )$ 
D)  $( - 1,1 , - 2 )$ 
A)  $( 5 , - 3 , - 2 )$ 
B)  $( - 1 , - 3 , - 1 )$ 
C)  $( - 1,1 , - 1 )$ 
D)  $( - 1,1 , - 2 )$

Question 7

Find the center and the radius of the sphere that has the given equation. $$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 8 y = 0$$

Accepted Answer

A)  $( 2 , - 4,0 ) , 2 \sqrt { 5 }$ 
B)  $( - 2,4,0 ) , 2 \sqrt { 5 }$ 
C)  $( - 2,4,0 ) , 20$ 
D)  $( 2 , - 4,0 ) , 20$ 
A)  $( 2 , - 4,0 ) , 2 \sqrt { 5 }$ 
B)  $( - 2,4,0 ) , 2 \sqrt { 5 }$ 
C)  $( - 2,4,0 ) , 20$ 
D)  $( 2 , - 4,0 ) , 20$

Question 8

Find an equation of the plane that passes through the line of intersection of the planes $x - z = 1$ and $y + 2 z = 3$, is perpendicular to the plane $x + y - 2 z = 2$.

Accepted Answer

The answer of Find an equation of the plane that...

Question 9

Which of the given lines is parallel to the line $x = 8 + t , y = t , z = - 7 - 10 t$ ?

Accepted Answer

A)  $x = - 5 + 3 t , y = 10 + t , z = - 2 - t$ 
B)  $x + 8 = y - 2 = - 9 - z$ 
C)  $x = - 5 t , y = 10 t , z = - 1 - t$ 
D)  $x = - 5 + t , y = 10 + t , z = - 1 - t$ 
E)  $r = ( - 6 , - 10,0 ) + t ( 2,2 , - 20 \}$ 
A)  $x = - 5 + 3 t , y = 10 + t , z = - 2 - t$ 
B)  $x + 8 = y - 2 = - 9 - z$ 
C)  $x = - 5 t , y = 10 t , z = - 1 - t$ 
D)  $x = - 5 + t , y = 10 + t , z = - 1 - t$ 
E)  $r = ( - 6 , - 10,0 ) + t ( 2,2 , - 20 \}$

Question 10

Find the distance between the planes. $$5 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 0$$

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 3 } \sqrt { 30 }$ 
C)  $\frac { 5 \sqrt { 30 } } { 6 }$ 
D)  $\frac { 5 } { 6 }$ 
E)  $\sqrt { 30 }$ 
A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 3 } \sqrt { 30 }$ 
C)  $\frac { 5 \sqrt { 30 } } { 6 }$ 
D)  $\frac { 5 } { 6 }$ 
E)  $\sqrt { 30 }$

Question 11

$\text { Find an equation for the surface obtained by rotating the line } x = 3 z \text { about the } x \text {-axis. }$

Accepted Answer

The answer of \(\text { Find an equation for the...

Question 12

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 3,0,5 )$ to the point $( 7,5,10 )$ along a straight line. The distance is measured in meters and the force in newtons.

Accepted Answer

A)  $64 \mathrm {~J}$ 
B)  $74 \mathrm {~J}$ 
C)  $84 \mathrm {~J}$ 
D)  $94 \mathrm {~J}$ 
E)  $104 \mathrm {~J}$ 
A)  $64 \mathrm {~J}$ 
B)  $74 \mathrm {~J}$ 
C)  $84 \mathrm {~J}$ 
D)  $94 \mathrm {~J}$ 
E)  $104 \mathrm {~J}$

Question 13

The tension $\boldsymbol { T }$ at each end of the chain has magnitude $27 \mathrm {~N}$ and makes an angle $\angle a = 27 ^ { \circ }$ with the horizontal. What is the weight of the chain? Round the result to the nearest hundredth.

Accepted Answer

The answer of The tension $\boldsymbol { T }$ at...

Question 14

Find the distance from $( - 10,2 , - 8 )$ to the $x y$-planes.

Accepted Answer

A)  8 
B)  16 
C)  10 
D)  2 
E)  4 
A)  8 
B)  16 
C)  10 
D)  2 
E)  4

Question 15

Find $| \mathbf { u } \times \mathbf { v } |$ correct to three decimal places where $| \mathbf { u } | = 18 , | \mathbf { v } | = 3 , \angle \theta = 80 ^ { \circ }$.

Accepted Answer

A)  $69.179$ 
B)  $69.179$. 
C)  $53.179$. 
D)  $75.179$. 
E)  $73.179$. 
A)  $69.179$ 
B)  $69.179$. 
C)  $53.179$. 
D)  $75.179$. 
E)  $73.179$.

Question 16

Determine whether the given points are collinear. $$A ( 2,3,2 ) , B ( - 4,0,5 ) , \text { and } C ( 4,4,1 )$$

Accepted Answer

A) Not collinear 
B) Collinear 
A) Not collinear 
B) Collinear

Question 17

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the $x z$-plane and 11 units to the right of it.

Accepted Answer

The answer of Write an inequality to describe the half-space...

Question 18

Find an equation for the surface obtained by rotating the parabola $y = 3 x ^ { 2 }$ about the $y$-axis.

Accepted Answer

A)  $z ^ { 2 } - 3 x ^ { 2 } - y ^ { 2 } = 1$ 
B)  $y ^ { 2 } + 3 x ^ { 2 } = z$ 
C)  $z ^ { 2 } - 3 x ^ { 2 } = y$ 
D)  $3 z ^ { 2 } + 3 x ^ { 2 } = y$ 
E)  $z ^ { 2 } + 3 x ^ { 2 } + y ^ { 2 } = 1$ 
A)  $z ^ { 2 } - 3 x ^ { 2 } - y ^ { 2 } = 1$ 
B)  $y ^ { 2 } + 3 x ^ { 2 } = z$ 
C)  $z ^ { 2 } - 3 x ^ { 2 } = y$ 
D)  $3 z ^ { 2 } + 3 x ^ { 2 } = y$ 
E)  $z ^ { 2 } + 3 x ^ { 2 } + y ^ { 2 } = 1$

Question 19

Find that the midpoint of the line segment from $P 1 ( - 20,20,3 )$ to $P 2 ( 6,24,7 )$.

Accepted Answer

A)  $( - 7 , - 22 , - 5 )$ 
B)  $( - 7,5,22 )$ 
C)  $( - 7,22,5 )$ 
D)  $( - 7,22 , - 5 )$ 
E)  $( - 7 , - 22,5 )$ 
A)  $( - 7 , - 22 , - 5 )$ 
B)  $( - 7,5,22 )$ 
C)  $( - 7,22,5 )$ 
D)  $( - 7,22 , - 5 )$ 
E)  $( - 7 , - 22,5 )$

Question 20

$\text { Find an equation of the sphere with center } ( 7 , - 5,7 ) \text { that touches the } x y \text {-plane. }$

Accepted Answer

A)  $( x + 7 ) ^ { 2 } + ( y - 5 ) ^ { 2 } + ( z + 7 ) ^ { 2 } = 49$ 
B)  $( x + 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$ 
C)  $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 40$ 
D)  $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$ 
E)  $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 7$ 
A)  $( x + 7 ) ^ { 2 } + ( y - 5 ) ^ { 2 } + ( z + 7 ) ^ { 2 } = 49$ 
B)  $( x + 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$ 
C)  $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 40$ 
D)  $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 49$ 
E)  $( x - 7 ) ^ { 2 } + ( y + 5 ) ^ { 2 } + ( z - 7 ) ^ { 2 } = 7$

Find an equation of the sphere with center $( 7 , - 5,7 )$ that touches the $x y$ -plane.

A woman walks due west on the deck of a ship at $6 \mathrm { mi } / \mathrm { h }$ . The ship is moving north at a speed of $20 \mathrm { mi } / \mathrm { h }$ . Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

Find an equation of the set of all points equidistant from the points $A ( 7 , - 8 , - 9 )$ and $B ( - 6,4 , - 12 )$ .

Let $\mathbf { v } = 7 \mathbf { j }$ and let $\mathbf { u }$ be a vector with length 5 that starts at the origin and rotates in the $x y$ - plane. Find the maximum value of the length of the vector $| \mathbf { u } \times \mathbf { v } |$ .

Find the standard equation of the sphere with center C and radius r. $C ( 4 , - 1,5 ) ; r = 8$

Find the midpoint of the line segment joining the given points. $( 4 , - 2 , - 3 )$ and $( 6 , - 4 , - 1 )$

Find the center and the radius of the sphere that has the given equation. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 8 y = 0$

Find an equation of the plane that passes through the line of intersection of the planes $x - z = 1$ and $y + 2 z = 3$ , is perpendicular to the plane $x + y - 2 z = 2$ .

Which of the given lines is parallel to the line $x = 8 + t , y = t , z = - 7 - 10 t$ ?

Find the distance between the planes. $5 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 0$

$\text { Find an equation for the surface obtained by rotating the line } x = 3 z \text { about the } x \text {-axis. }$

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 3,0,5 )$ to the point $( 7,5,10 )$ along a straight line. The distance is measured in meters and the force in newtons.

Find the distance from $( - 10,2 , - 8 )$ to the $x y$ -planes.

Find $| \mathbf { u } \times \mathbf { v } |$ correct to three decimal places where $| \mathbf { u } | = 18 , | \mathbf { v } | = 3 , \angle \theta = 80 ^ { \circ }$ .

Determine whether the given points are collinear. $A ( 2,3,2 ) , B ( - 4,0,5 ) , \text { and } C ( 4,4,1 )$

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the $x z$ -plane and 11 units to the right of it.

Find an equation for the surface obtained by rotating the parabola $y = 3 x ^ { 2 }$ about the $y$ -axis.

Find that the midpoint of the line segment from $P 1 ( - 20,20,3 )$ to $P 2 ( 6,24,7 )$ .

$\text { Find an equation of the sphere with center } ( 7 , - 5,7 ) \text { that touches the } x y \text {-plane. }$

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

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Multiple Integrals

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Filters

Exam 12: Vectors and the Geometry of Space

Find an equation of the sphere with center (7,−5,7)( 7 , - 5,7 )(7,−5,7) that touches the xyx yxy -plane.

A woman walks due west on the deck of a ship at 6mi/h6 \mathrm { mi } / \mathrm { h }6mi/h . The ship is moving north at a speed of 20mi/h20 \mathrm { mi } / \mathrm { h }20mi/h . Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

Find an equation of the set of all points equidistant from the points A(7,−8,−9)A ( 7 , - 8 , - 9 )A(7,−8,−9) and B(−6,4,−12)B ( - 6,4 , - 12 )B(−6,4,−12) .

Let v=7j\mathbf { v } = 7 \mathbf { j }v=7j and let u\mathbf { u }u be a vector with length 5 that starts at the origin and rotates in the xyx yxy - plane. Find the maximum value of the length of the vector ∣u×v∣| \mathbf { u } \times \mathbf { v } |∣u×v∣ .

Find the standard equation of the sphere with center C and radius r. C(4,−1,5);r=8C ( 4 , - 1,5 ) ; r = 8C(4,−1,5);r=8

Find the midpoint of the line segment joining the given points. (4,−2,−3)( 4 , - 2 , - 3 )(4,−2,−3) and (6,−4,−1)( 6 , - 4 , - 1 )(6,−4,−1)

Find the center and the radius of the sphere that has the given equation. x2+y2+z2−4x+8y=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 8 y = 0x2+y2+z2−4x+8y=0

Find an equation of the plane that passes through the line of intersection of the planes x−z=1x - z = 1x−z=1 and y+2z=3y + 2 z = 3y+2z=3 , is perpendicular to the plane x+y−2z=2x + y - 2 z = 2x+y−2z=2 .

Which of the given lines is parallel to the line x=8+t,y=t,z=−7−10tx = 8 + t , y = t , z = - 7 - 10 tx=8+t,y=t,z=−7−10t ?

Find the distance between the planes. 5x−2y+z−4=0,5x−2y+z+6=05 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 05x−2y+z−4=0,5x−2y+z+6=0

Find an equation for the surface obtained by rotating the line x=3z about the x-axis. \text { Find an equation for the surface obtained by rotating the line } x = 3 z \text { about the } x \text {-axis. } Find an equation for the surface obtained by rotating the line x=3z about the x-axis.

Find the work done by a force F=i+5j+7k\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }F=i+5j+7k that moves an object from the point (3,0,5)( 3,0,5 )(3,0,5) to the point (7,5,10)( 7,5,10 )(7,5,10) along a straight line. The distance is measured in meters and the force in newtons.

The tension T\boldsymbol { T }T at each end of the chain has magnitude 27 N27 \mathrm {~N}27 N and makes an angle ∠a=27∘\angle a = 27 ^ { \circ }∠a=27∘ with the horizontal. What is the weight of the chain? Round the result to the nearest hundredth.

Find the distance from (−10,2,−8)( - 10,2 , - 8 )(−10,2,−8) to the xyx yxy -planes.

Find ∣u×v∣| \mathbf { u } \times \mathbf { v } |∣u×v∣ correct to three decimal places where ∣u∣=18,∣v∣=3,∠θ=80∘| \mathbf { u } | = 18 , | \mathbf { v } | = 3 , \angle \theta = 80 ^ { \circ }∣u∣=18,∣v∣=3,∠θ=80∘ .

Determine whether the given points are collinear. A(2,3,2),B(−4,0,5), and C(4,4,1)A ( 2,3,2 ) , B ( - 4,0,5 ) , \text { and } C ( 4,4,1 )A(2,3,2),B(−4,0,5), and C(4,4,1)

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the xzx zxz -plane and 11 units to the right of it.

Find an equation for the surface obtained by rotating the parabola y=3x2y = 3 x ^ { 2 }y=3x2 about the yyy -axis.

Find that the midpoint of the line segment from P1(−20,20,3)P 1 ( - 20,20,3 )P1(−20,20,3) to P2(6,24,7)P 2 ( 6,24,7 )P2(6,24,7) .

Find an equation of the sphere with center (7,−5,7) that touches the xy-plane. \text { Find an equation of the sphere with center } ( 7 , - 5,7 ) \text { that touches the } x y \text {-plane. } Find an equation of the sphere with center (7,−5,7) that touches the xy-plane.

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Find an equation of the sphere with center $( 7 , - 5,7 )$ that touches the $x y$ -plane.

A woman walks due west on the deck of a ship at $6 \mathrm { mi } / \mathrm { h }$ . The ship is moving north at a speed of $20 \mathrm { mi } / \mathrm { h }$ . Find the speed of the woman relative to the surface of the water. Round the result to the nearest tenth.

Find an equation of the set of all points equidistant from the points $A ( 7 , - 8 , - 9 )$ and $B ( - 6,4 , - 12 )$ .

Let $\mathbf { v } = 7 \mathbf { j }$ and let $\mathbf { u }$ be a vector with length 5 that starts at the origin and rotates in the $x y$ - plane. Find the maximum value of the length of the vector $| \mathbf { u } \times \mathbf { v } |$ .

Find the standard equation of the sphere with center C and radius r. $C ( 4 , - 1,5 ) ; r = 8$

Find the midpoint of the line segment joining the given points. $( 4 , - 2 , - 3 )$ and $( 6 , - 4 , - 1 )$

Find the center and the radius of the sphere that has the given equation. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 8 y = 0$

Find an equation of the plane that passes through the line of intersection of the planes $x - z = 1$ and $y + 2 z = 3$ , is perpendicular to the plane $x + y - 2 z = 2$ .

Which of the given lines is parallel to the line $x = 8 + t , y = t , z = - 7 - 10 t$ ?

Find the distance between the planes. $5 x - 2 y + z - 4 = 0,5 x - 2 y + z + 6 = 0$

$\text { Find an equation for the surface obtained by rotating the line } x = 3 z \text { about the } x \text {-axis. }$

Find the work done by a force $\mathbf { F } = \mathbf { i } + 5 \mathbf { j } + 7 \mathbf { k }$ that moves an object from the point $( 3,0,5 )$ to the point $( 7,5,10 )$ along a straight line. The distance is measured in meters and the force in newtons.

Find the distance from $( - 10,2 , - 8 )$ to the $x y$ -planes.

Find $| \mathbf { u } \times \mathbf { v } |$ correct to three decimal places where $| \mathbf { u } | = 18 , | \mathbf { v } | = 3 , \angle \theta = 80 ^ { \circ }$ .

Determine whether the given points are collinear. $A ( 2,3,2 ) , B ( - 4,0,5 ) , \text { and } C ( 4,4,1 )$

Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to the $x z$ -plane and 11 units to the right of it.

Find an equation for the surface obtained by rotating the parabola $y = 3 x ^ { 2 }$ about the $y$ -axis.

Find that the midpoint of the line segment from $P 1 ( - 20,20,3 )$ to $P 2 ( 6,24,7 )$ .

$\text { Find an equation of the sphere with center } ( 7 , - 5,7 ) \text { that touches the } x y \text {-plane. }$