Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)
The volume of air contained in the model is 66 cm³ (or 21π cm³).
Explanation
This is a combined solid problem from Surface Areas and Volumes. The model consists of a cylinder with two cones attached at its ends. The total volume is the sum of the volume of the cylinder and the volumes of both cones. Since the aluminium sheet is thin, the inner and outer dimensions are nearly the same, so we calculate the volume directly.
Solution Steps
Step 1: Find the radius and height of each component. Diameter = 3 cm, so radius (r) = 1.5 cm. Total length of model = 12 cm. Height of each cone = 2 cm. Since there are two cones, total height of both cones = 2 + 2 = 4 cm. Therefore, height of cylinder (h) = 12 - 4 = 8 cm.
Step 2: Calculate the volume of the cylinder. Volume of cylinder = πr²h = π × (1.5)² × 8 = π × = 18π cm³
Step 3: Calculate the volume of both cones. Volume of one cone = (1/3)πr²h = (1/3) × π × (1.5)² × 2 = (1/3) × π × = 1.5π cm³ Volume of two cones = π = 3π cm³
Step 4: Find the total volume. Total volume = Volume of cylinder + Volume of two cones = 18π + 3π = 21π cm³ Using π = 22/7: = 21 × (22/7) = = 66 cm³