a rock is thrown into a still pond. the circular ripples move outward from the point of impact of the rock…

a rock is thrown into a still pond. the circular ripples move outward from the point of impact of the rock so that the radius of the circle formed by a ripple increases at the rate of 4 feet per second. find the rate at which the area is changing at the instant the radius is 15 feet. when the radius is 15 feet, the area is changing at approximately 120π square feet per second. (round to the nearest thousandth as needed.)

a rock is thrown into a still pond. the circular ripples move outward from the point of impact of the rock so that the radius of the circle formed by a ripple increases at the rate of 4 feet per second. find the rate at which the area is changing at the instant the radius is 15 feet. when the radius is 15 feet, the area is changing at approximately 120π square feet per second. (round to the nearest thousandth as needed.)

Answer

Answer:

$120\pi$

Explanation:

Step1: Recall area formula for circle

$A = \pi r^{2}$

Step2: Differentiate with respect to time $t$

$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$

Step3: Identify given values

$r = 15$ feet, $\frac{dr}{dt}=4$ feet per second

Step4: Substitute values

$\frac{dA}{dt}=2\pi\times15\times4 = 120\pi$ square - feet per second