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%
