Apentix/AP Calculus AB/Unit 4: Contextual Applications of Differentiation/Question
AP Calculus ABhardmcq1 pt

A pebble is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a constant rate of 22 feet per second. Which of the following correctly relates the rate of change of the area AA enclosed by the ripple to its radius rr?

A.dAdt=2πrdrdt\frac{dA}{dt} = 2\pi r \frac{dr}{dt}
B.dAdt=πr2drdt\frac{dA}{dt} = \pi r^2 \frac{dr}{dt}
C.dAdt=4πrdrdt\frac{dA}{dt} = 4\pi r \frac{dr}{dt}
D.A=πr2drdtA = \pi r^2 \frac{dr}{dt}

Explanation

Core Concept

Correct. The area of a circle is A=πr2A = \pi r^2. Differentiating both sides with respect to time tt yields dAdt=2πrdrdt\frac{dA}{dt} = 2\pi r \frac{dr}{dt}.

Correct Answer

AdAdt=2πrdrdt\frac{dA}{dt} = 2\pi r \frac{dr}{dt}

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