If a snowball melts so that its surface area decreases at a rate of 7 cm2/min, find the rate (in cm/min) at which the diameter decreases when the diameter is 11 cm

Answer 1

If a snowball melts so that its surface area decreases at a rate of 7 cm2/min, find the rate (in cm/min) at which the diameter decreases when the diameter is 11 cm

Hello Nick B.,

Surface Area of a sphere:

S = 4 π r²

dS/dt = - 7cm²/min

Find rate of decrease of Diameter:

dD/dt = ?

Substitute r in the equation with radius = 1/2*Diameter:

r = 1/2D

Take the derivative of the surface area function with respect to time:

d/dt [S = 4 π (1/2D)²]

d/dt [S = 4 π 1/4*D²]

d/dt [S = π *D²]

dS/dt = 2πD*dD/dt

Solve for dD/dt:

dD/dt = dS/dt / 2πD

Substitute, dS/dt = - 7cm²/min, and D = 11 cm

dD/dt = dS/dt / 2πD

dD/dt = - 7cm²/min / 2π*11cm

dD/dt = - 7cm²/min / 22π cm

dD/dt = - 7cm / 22π min

dD/dt = - 0.101 cm/min