Consider a circle whose radius varies over time

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

Consider a circle whose radius varies over time. At instant

t_1

the radius is 5 cm and is increasing at a rate of 0.40 cm s⁻¹. At a later instant

t_2

the radius is again 5 cm but is now decreasing at a rate of 0.40 cm s⁻¹. (a) Without performing any numerical differentiation, determine the sign (positive, negative, or zero) of the instantaneous rate of change of the circle’s area

\dfrac{dA}{dt}

at each instant

t_1

and

t_2

. Explain your reasoning. (b) Explain why the magnitude of

\dfrac{dA}{dt}

is the same at

t_1

and

t_2

even though the signs differ. Reference the relevant geometric relationship in your explanation. (c) Suppose instead that the radius were 10 cm at both instants while

\dfrac{dr}{dt}

remained

\pm0.40\text{ cm s}^{-1}

as before. Describe qualitatively (no numbers necessary) how the magnitude of

\dfrac{dA}{dt}

would compare to the magnitudes found in parts (a) and (b), and justify your conclusion.

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