SOLUTION: A sphere has a radius of 24 cm. A rectangular prism has sides of (a), (a/2), (a/4). a. Determine (a) of a prism that has the same volume as the sphere. b. Determine (a) of a

Question 896848: A sphere has a radius of 24 cm. A rectangular prism has sides of (a), (a/2), (a/4).
a. Determine (a) of a prism that has the same volume as the sphere.
b. Determine (a) of a prism that has the same surface area as the sphere.
I found the value of the sphere and tried to find the value of the prism, to subtract the prism from the sphere, and then determine the diameter of the prism to find (a). However, I couldn't go only having the radius and volume of the sphere.
Please I need some help. Thank you very much in advance.
Found 2 solutions by josgarithmetic, Theo:
Answer by josgarithmetic(39618) (Show Source): You can put this solution on YOUR website!
Question (a):

volume of sphere,
volume of rectangular prism, in your case, .

You are given r=24 for the sphere,and that you want to find a for the prism being also this same volume.

Solve for a.

Try to do question (b) yourself. Make the proper expression for the surface area of your prism; find the formula for surface area of a sphere in your book or other good reference material.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you only need the radius of the sphere.

you can figure everything else out from that.

you have a sphere with a radius of 24 cm.

volume of the sphere is equal to 4/3*pi*r^3 = 57905.8357...

surface area of the sphere is equal to 4*pi*r^2 = 7328.2294...

the volume of the prism is equal to a * a/2 * a/4 which is equal to a^3 / 8

since the volume of the prism must be equal to the volume of the sphere, you get:

a^3 / 8 = 57905.8357...

solve for a to get:

a = 77.3756...

that's your first solution.

a * a/2 * a/4 a^3 / 8 = 57905.8357... which is the volume of the sphere.

the surface area of your rectangular prism is going to be:

let's assume the height of your prism is a and the length of your prism is a/2 and the width of your prism is a/4

you have:

height = a
length = a/2
width = a/4

you will have 2 faces that are height times length.
you will have 2 faces that are height times width.
you will have 2 faces that are length times width.

your surface area will then be equal to:

2 * height * length = 2 * a * a/2
2 * height * width = 2 * a * a/4
2 * length * width = 2 * a/2 * a/4

simplify these to get:

2 * height * length = a^2
2 * height * width = a^2/2
2 * length * width = a^2/4

add up the surface area of the prism and you get;

SA = a^2 + a^2/2 + a^2/4 which becomes:

SA = 4a^2/4 + 2a^2/4 + a^2/4 which becomes:

SA = 7a^2/4

since the surface area of the sphere is equal to 7238.2294..., and the surface area of the prism needs to be equal to that, you get:

7a^2/4 = 7238.229474...

solve for a to get a = 64.3127...

in order for the volume of the prism to be equal to the volume of the sphere, a must be equal to 77.3756...

when a is equal to 77.3756...
a/2 = 38.6878...
a/4 = 19.3439...

a * a/2 * a/4 becomes equal to 57905.8357... which is equal to the volume of the sphere.

in order for the surface area of the prism to be equal to the surface area of the sphere, a must be equal to 64.3127...

when a is equal to 64.3127...
a/2 = 32.1563...
a/4 = 16.0781...

surface area of the prism is equal to 2*a*a/2 + 2*a*a/4 + 2*a/2*a/4 = 7238.229474... which is equal to the surface area of the sphere.

looks like we did good.

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