Question

Three balls are packaged in a cylindrical container as shown below. The balls just touch the top, bottom, and sides of the cylinder. The diameter of each ball is 13 cm.
a. What is the volume of the cylinder? Explain your method for finding the volume.
b. What is the total volume of the three balls? Explain your method for finding the total volume.
c. What percent of the volume of the container is occupied by the three balls? Explain how you would find the percent.

Answer 1

A. The volume of the cylinder can be computed because the statement "The balls just touch the top, bottom, and sides of the cylinder" gives a clue of the height and radius of the cylinder. The height is equal to 3 times the diameter of the balls, and its radius is the same as the balls' radius.

Vcylinder = πr^2h = π(13/2)^2 (3*13) = 5176.56 cm^3

B. The volume of the three balls is computed using the equation of V for spheres:

Vball = 3*4*π*r^2/3 = 3451.05 cm^3

C. The answer is the equal to the total volume of the balls divided by the volume of the cylinder

3451.05(100%)/5176.56 = 66.67%

Answer 2

We know the following:
Cylinder volume: V₁ = π r² h
Ball (sphere) volume:V₂ = $\frac{4}{3}$ π r³
where:
V - volume
r - radius of base of cylinder and diameter of ball
h - height of cylinder.

R = 13 cm ⇒ r = 13 ÷ 2 = 6.5
π = 3.14

a) Since balls touch all sides of cylinder (as shown in image), it can be concluded that height of cylinder is equal to sum of diameters of 3 balls and that radius of base of cylinder is equal to radius of ball:
h = 3 × r = 3 × 13 cm = 39 cm
r = 6.5 cm
So,
V₁ = π r² h
V₁ = 3.14 × (6.5 cm)² × 39 cm
V₁ = 5,173.9 cm³

b. The total volume of three balls is the sum of volumes of each ball:
Vₐ = 3 × V₂ 
Vₐ = 3 × $\frac{4}{3}$ π r³
Vₐ = 3 × $\frac{4}{3}$ 3.14 (6.5 cm)³
Vₐ = 3,449.3 cm³

c. Percentage of the volume of the container occupied by three balls ould be expressed as ratio of volume of three balls and volume of cylinder:
V = $\frac{ V_{1}}{ V_{a} }$ ×100 
V = $\frac{3,449.3}{5,173.9}$ ×100
V = 0.6666 ×100
V = 66.66%