If the radius of a right circular cone is increased by p% without increasing its height, then what is the percentage increase in the volume of the cone?a)p2b)2p2c)p2/100d)p(2 + p/100)Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Solution:

Let's assume the original radius of the cone is 'r' and the original height is 'h'.

The volume of a cone is given by the formula:

V = (1/3) * π * r^2 * h

If the radius is increased by p%, then the new radius would be (r + (p/100)r) = (1 + p/100)r.

Since the height is not changing, the new volume of the cone would be:

V' = (1/3) * π * (1 + p/100)^2 * r^2 * h

To find the percentage increase in volume, we need to calculate the difference between the new volume and the original volume, and then express it as a percentage of the original volume.

Percentage increase in volume = ((V' - V)/V) * 100

Let's calculate this step by step.

Step 1:
V' = (1/3) * π * (1 + p/100)^2 * r^2 * h

Step 2:
V' - V = [(1/3) * π * (1 + p/100)^2 * r^2 * h] - [(1/3) * π * r^2 * h]
= (1/3) * π * r^2 * h * [(1 + p/100)^2 - 1]

Step 3:
V' - V = (1/3) * π * r^2 * h * [1 + 2(p/100) + (p/100)^2 - 1]
= (1/3) * π * r^2 * h * (2(p/100) + (p/100)^2)

Step 4:
(V' - V)/V = [(1/3) * π * r^2 * h * (2(p/100) + (p/100)^2)] / [(1/3) * π * r^2 * h]
= (2(p/100) + (p/100)^2)

Step 5:
Percentage increase in volume = ((V' - V)/V) * 100
= [(2(p/100) + (p/100)^2)] * 100
= 2p + (p^2/100)

Therefore, the percentage increase in the volume of the cone is p(2 + p/100), which is represented by option 'D'.

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