The radius of a sphere is increasing at a rate of 2 cm/s. Find the rate at which the volume is increasing when the radius is 5 cm.

The volume is increasing at a rate of 628.32 cm³/s when r = 5 cm.

Step-by-Step Solution:
Volume formula: V = (4/3)πr³
Using chain rule: dV/dt = (4/3)π × 3r² × dr/dt
Simplify: dV/dt = 4πr² × dr/dt
Substitute r = 5 cm and dr/dt = 2 cm/s:
dV/dt = 4π(5)² × 2 = 628.32 cm³/s

Key Mistake to Avoid: Students often forget to multiply by dr/dt after differentiating with respect to r, missing the chain rule application.



Quick Tip: Always write down the chain rule formula dV/dt = dV/dr × dr/dt before starting related rates problems to avoid missing steps.