Question

The base diameter and altitude of a right circular cone are observed at a certain instant to be 10 and 20 inches, respectively. If the lateral area is constant and the base diameter is increasing at a rate of 1 inch per minute, find the rate at which the altitude is decreasing.

Answer 1

Given: d = 10, h = 20, S - is constant.

x = 1 (inch/minute) - the rate at which the diameter is increasing.

Find: $y$ (inch/minute) - the rate at which the altitude is decreasing.

S₀ = d * $\frac{h}{2}$

S₀ = 10 * $\frac{20}{2}$ = 100.

S₁ - the lateral area in one minute.

d₁ = d + x

d₁ = 10 + 1

d₁ = 11.

$h$₁ = $h$ - $y$

$h$₁ = 20 - $y$.

S₁ = d₁ * $\frac{h1}{2}$

S₁ = 11 * ($\frac{20 - y}{2}$),

By condition, S₁ = S₀.

11 * ($\frac{20 - y}{2}$) = 100

11 * (20 - $y$) = 100 * 2.

20 - $y$ = 200 / 11.

$-y$ = 18$\frac{2}{11}$ - 20.

$-y$ = $-1\frac{9}{11}$

$y$ = $1\frac{9}{11}$.

Answer: $1\frac{9}{11}$ inch per minute.