Question

The radius of a cylinder is increasing at the rate 2 cm/sec and its height is decreasing at the rate 3 cm/sec, then find the rate of change of volume when the radius is 3cm and the height is 5 cm.

Answer 1

The volume of a cylinder is given by the formula V = πr^2h,

where r is the radius and h is the height.

We are given that the radius is increasing at the rate of 2 cm/sec, which means dr/dt = 2 cm/sec, and that the height is decreasing at the rate of 3 cm/sec, which means dh/dt = -3 cm/sec.

We want to find the rate of change of volume when the radius is 3 cm and the height is 5 cm.

So, we need to find dV/dt when r = 3 cm and h = 5 cm.

Using the product rule of differentiation, we can write: dV/dt = π(2rh)(dr/dt) + π(r^2)(dh/dt)

Substituting the given values, we get: dV/dt = π(2 x 3 x 5)(2) + π(3^2)(-3) dV/dt = 30π - 27π dV/dt = 3π

Therefore, the rate of change of volume when the radius is 3cm and the height is 5 cm is 3π cubic cm/sec.