The lateral surface area of the cone is 36pi cm2. What should be the length

The lateral surface area of the cone is 36pi cm2. What should be the length of the radius of its base for the volume of the cone to be the largest?

To calculate what the length of the radius of the base of the cone should be so that the volume of the cone is the largest, we will start by recalling the formula for calculating the area of the lateral surface:

Sside = π * R * L;

Let us express L:

π * R * L = 36π;

R * L = 36;

L = 36 / R,

where L is the length of the generatrix, R is the radius of the base.

We find the volume of the cone by the formula:

V = 1/3 * π * R ^ 2 * H = 1/3 * πR ^ 2 * √ (L ^ 2 – R ^ 2) = 1/3 * πR ^ 2√ ((36 / R) ^ 2 – R ^ 2) = 1 / 3π√ (1296R ^ 2 – R ^ 4).

We introduce the replacement: x = R ^ 2.

V = π / 3 * √ (1296x – x ^ 2) = π / 3 * √ (648 ^ 2 – (x ^ 2 – 2x * 648 + 648 ^ 2)) = π / 3√ (648 ^ 2 – ( x – 648) ^ 2).

V = Vmax, if x = 648, therefore R ^ 2 = 648; R ^ 2 = 324 * 2; R = 18√2 cm.