David has two cylinder shaped bottles of punch, each with a height of 12 inches and a radius of 2 inches. He wants to fill a half-sphere punch bowl to the brim with the punch. What should the radius of the punch bowl be to the nearest tenth of an inch to fill it entirely with the two bottles of punch?

Answer: 5.2 inches

Step-by-step explanation:

Hi, first we have to find the volume of the cylinder shaped bottles:

V = π r^2 h

where:

V = volume

r = radius

h = height

Replacing with the values given:

V = π 2^2 (12) = 150.80 cubic inches

Since there are 2 bottles

Total punch: 2V = 301.59 cubic inches

The formula for the volume of a half-sphere is

V = (2 / 3) π r3.

Since the half-sphere is filled with the two bottles of punch (301.59 cubic inches)

301.59 = (2 / 3) π r3.

301.59 / (2/3) π = r3

144 = r3

∛144 = r

5.2 inches = r