Question

the volume v of a melting snowball is decreasing at at rate of 4 cm³ per second. let the variable t represent the time, in seconds, since we started our investigation. find the rate at which the radius of the snowball is decreasing with respect to time at the instant when the radius of the snow ball is 3 .

Answer 1

The rate at which the radius of the snowball is decreasing with respect to time is approximately 0.035 cm/s when the radius of the snowball is 3 cm.

What is the rate at which the radius is decreasing?

The formula for the volume of the snowball to its radius r is given by the formula:

V = ⁴/₃πr³

Take the derivative of both sides with respect to time t, we get:

dV/dt = 4πr²(dr/dt)

where dr/dt is the rate at which the radius is changing with respect to time.

dV/dt = -4 cm³/s (negative because the volume is decreasing),

To find dr/dt when the radius is 3 cm.

substitute these values and solve for dr/dt:

-4 = 4π(3)²(dr/dt)

Solving for dr/dt gives:

dr/dt = -0.035 cm/s

Thus, the rate at which the radius of the snowball is decreasing with respect to time is approximately 0.035 cm/s when the radius of the snowball is 3 cm.