Volume

13.
(a)
The circular frustum is made by cutting off a smaller cone from a larger cone.
The diameter and height of the larger cone are 16 cm and x cm respectively.

Diameter of the smaller cone
= Diameter of the cylinder
= 8 cm

The larger cone and smaller cone are similar in shape.
(x cm) : (The height of the smaller cone) = (16 cm) : (8 cm)
Height of the smaller cone = 8x/16 cm = x/2 cm

Height of the circular frustum
= [x - (x/2)] cm
= x/2 cm


(b)
Volume of the cylinder
= (Base area) * Height
= [π * (8/2)²] * [20 - (x/2)] cm³
= (320π - 8πx) cm³


(c)
Volume of the circular frustum
= (Volume of the large cone) - (Volume of the smaller cone)
= (1/3) * π * (16/2)² * x - (1/3) * π * (8/2)² * (x/2)cm³
= (64/3)πx - (8/3)πx cm³
= (56/3)πx cm³

Volume of container ≥ 2L = 2000 cm³
320π - 8πx + (56/3)πx ≥ 2000
(32/3)πx ≥ 2000 - 320π
πx ≥ 187.5 - 30π
x ≥ (187.5 - 30π) / π