A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?
The enclosed area is increasing at the rate of 80π cm²/s.
Explanation
This question is identical to Example 3 in the textbook, with different numerical values. The solution method uses the chain rule to find the rate of change of area with respect to time. The key insight is that area A = πr², and by chain rule, dA/dt = 2πr × dr/dt. Substituting the given values r = 8 cm and dr/dt = 5 cm/s yields the answer.
Solution Steps
Step 1: Let A be the area of the circular wave. The area of a circle with radius r is given by A = πr².
Step 2: To find the rate of change of area with respect to time, apply the Chain Rule: dA/dt = d/dr(πr²) × dr/dt = 2πr × dr/dt.
Step 3: It is given that dr/dt = 5 cm/s (speed of the wave) and r = 8 cm.
Step 4: Substitute the values: dA/dt = 2π × = 80π cm²/s.
Step 5: Thus, the enclosed area is increasing at the rate of 80π cm²/s when the radius is 8 cm.