Question

A potter forms a 100 cm3 piece of clay into a cylinder. As she rolls it, the length L,of the cylinder increases and the radius, r, decreases. If the length of the cylinder is increasing at 0.1 cm per second, find the rate at which the radius is changing when the radius is 5 cm.

Answer 1

- 0.2 cm/s

Explanation:

Volume of a cylinder = πr²l--------------------------------------------------------(1)

dV/dt =(dV /dr ) x (dr/dt) + (dV /dl ) x (dl/dt) ---------------------------------(2)

dV/dr = 2 πrl

dV/dl = πr²

dr/dt = Unknown

dl/dt = 0.1 cm/s

dV/dt = 0

From equation (1), the length of the cylinder can be calculated when r = 5cm

V = πr²L

100 = π (5)²L

L =100/25π

=4/π

To find the rate of radius change (dr/dt) we substitute known values into equation (2):

0 = 2 π (5) (4/π) x (dr/dt) + π(5)² x 0.1

0 = 40 (dr/dt) + 2.5π

40 (dr/dt) = -2.5π

dr/dt = -2.5π/40

= -0.1963 cm/s

≈ - 0.2 cm/s

The negative sign shows that the radius of the cylinder of constant volume decrease at a rate twice the length.