1. The volume of a sphere is numerically equal to half its surface area. What is the radius of the sphere?

2. A plane cuts through a sphere with diameter 20 cm, but the closest it gets to the center is 3 cm. What is the area of the intersection of the sphere and the plane in sq cm?

3. The surface area of a sphere is 36pi. Find the volume of the sphere.

4. All eight vertices of a unit cube are on a sphere (i.e. the cube is inscribed in the sphere). What is the surface area of the sphere?

5. The surface area of a sphere is 1. What is the surface area (including the base area) of a hemisphere with the same radius? View image: https://latex.artofproblemsolving.com/3/7/0/370b8ce4d4b2db15f747a870a9cc97a946e120a3.png

6. The surface area of planet AoPS is 100 times that of Earth, and its volume is times that of Earth. What is n. (Assume both planets are perfect spheres.)

7. The two bases of a cylinder are two parallel cross sections of a sphere. We know the radius of the sphere is 3 and the height of the cylinder is 4. Find the volume of the cylinder. View image: https://latex.artofproblemsolving.com/0/e/4/0e477889f2d42dfaf17c33b488d1a19527992549.png

8. Sphere S is tangent to all 12 edges of a cube with edge length 6. Find the volume of the sphere.

9. A and B are two points on a unit sphere. We know the space distance between A and B is sqrt(2). What is the length of shortest path on the sphere that connects A to B.

10. As shown in the diagram, a hemisphere sits on top of a cylinder with the same radius. We know the area of the bottom base of the cylinder is 1/6th of the surface area of the combined shape. What fraction of the volume of the combined shape is the volume of the cylinder? View image: https://latex.artofproblemsolving.com/0/6/9/06970ee085d9b84503a4bf2734376b74845fb985.png

11. Points A,B, and C are on a sphere whose radius is 13. If AB=BC=10, what is the longest possible value of AC?

web2.0calcmember Apr 12, 2019

#1**0 **

1 - Volume =4/3 x pi x r^3

SA =4 x pi x r^2

4/3*pi*r^3 =1/2[4*pi*r^2], solve for r

**Radius = 3/2**

Guest Apr 12, 2019

#2**+3 **

2. A plane cuts through a sphere with diameter 20 cm, but the closest it gets to the center is 3 cm. What is the area of the intersection of the sphere and the plane in sq cm?

Looking at a cross-section of thec sphere....we have a great circle with a dadius of 10 units

Let the equation of this circle be x^2 + y^2 = 100

The intersection of the plane and sphere will create a smaller circle

Let y = 3 and we can solve for x^2 = the radius of the smaller circle

x^2 + (3)^2 = 100

x^2 = 100 - 9

x^2 = 91 = r^2

So....the area of this smaller circle = pi * r^2 = 91 pi cm^2

CPhill Apr 12, 2019

#3**+3 **

3. The surface area of a sphere is 36pi. Find the volume of the sphere.

SAsphere = 4 pi ^ r^2

36 pi = 4 pi * r^2

36 = 4r^2

9 = r^2

3 = r

So.....

Vsphere = (4/3) pi * r^3 = (4/3) pi (3)^3 = 4 pi (3)^2 = 36 pi units^3

CPhill Apr 12, 2019

#4**+3 **

4. All eight vertices of a unit cube are on a sphere (i.e. the cube is inscribed in the sphere). What is the surface area of the sphere?

The long diagonal of the cube will = sqrt (3) units

The radius of the sphere is 1/2 of this

So....the surface area of the sphere =

4 pi [ radius]^2 =

4 pi [ sqrt (3) / 2 ] ^2 =

4 pi (3/4) =

3 pi units^2

CPhill Apr 12, 2019

#5**+3 **

5. The surface area of a sphere is 1. What is the surface area (including the base area) of a hemisphere with the same radius?

SA _{sphere} = 4 pi r^2

1 = 4pi * r^2

1 /[ 4pi ] = r^2

Area of base = pi * r^2 = pi * (1) /( 4 pi ) = (1/4) units^2

So....surface area = (1/2) of sphere surface area + area of base = (1/2) +(1/4) = 3/4 units^2

CPhill Apr 12, 2019

#6**+3 **

7. The two bases of a cylinder are two parallel cross sections of a sphere. We know the radius of the sphere is 3 and the height of the cylinder is 4. Find the volume of the cylinder.

Considering a cross-section.....

The radius of the sphere will be the hyptonuse of a right triangle and 1/2 the height of the cylinder will be one of the legs

The radius^2 of the cylinder =

r^2 = 3^2 - 2^2 = 5

So....the volume of the cylinder =

pi * r^2 * h =

pi * 5 * 4 =

20 pi units^3

CPhill Apr 12, 2019

#7**+3 **

8. Sphere S is tangent to all 12 edges of a cube with edge length 6. Find the volume of the sphere.

The radius of the sphere = (1/2) 6 √2 = 3 √2 units

So....the volume of the sphere =

(4/3)pi ( 3√2)^3 =

(4/3)pi (27)* 2√2 =

72√2 pi units^3

CPhill Apr 12, 2019

#8**+3 **

9. A and B are two points on a unit sphere. We know the space distance between A and B is sqrt(2). What is the length of shortest path on the sphere that connects A to B.

Considering a cross-section....the space distance AB will serve as the hypoenuse of a right triangle will legs of 1

So....the central angle separating A and B wil = 90°

So....the shortest distance between A and B on the sphere wiil = 2 pi (r) * (90/360)= 2 ( 1/4) pi = pi /2 units

CPhill Apr 12, 2019

#11**+2 **

10. As shown in the diagram, a hemisphere sits on top of a cylinder with the same radius. We know the area of the bottom base of the cylinder is 1/6th of the surface area of the combined shape. What fraction of the volume of the combined shape is the volume of the cylinder?

The combined surface area = base area of cylinder + lateral surface area of cylinder + surface area of hemisphere

(1/6)combined surface area = [ pi* r^2 + 2 pi * r * h + 2 pi r^2] / 6 = ( 3 pi r^2 + 2 pi *r * h) / 6

Area of bottom base of cylinder = pi * r^2

So

pi r^2 =( 3pi r^2 + 2pi * r *h) / 6

6 pi r^2 = 3pi r^2 + 2pi *r * h

3pi r^2 = 2pi r h

(3/2) r = h = height of cylinder

So....volume of cylinder =

pi * r^2 * (3/2)r = (3/2) pi r^3 (1)

Vol of combined shape = volume of cylinder + volume of hemisphere = (3/2)pi r^3 + (2/3)pi *r^3 =

(13/6) pi r^3 (2)

So ratio of (1) to (2) =

(3/2) / ( 13/6) =

(3/2) * (6/13) =

18 /26 =

9 : 13

CPhill Apr 12, 2019