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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

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

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

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

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  

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

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

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

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