A thin layer of oil spreads on the surface of a calm lake

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

Problem

A thin layer of oil spreads on the surface of a calm lake. The radius of the circular spill at time

t

minutes after the leak begins is given by

r(t)=5\sqrt{t+1}.

The area of the spill is

A(t)=\pi r(t)^2

. Assume the model is valid for

t\ge 0

. (a) Calculate

\dfrac{dA}{dt}

as an explicit function of

t

. State the units of

\dfrac{dA}{dt}

. (b) Determine whether the area of the spill is increasing or decreasing for all

t\ge 0

. Explain your conclusion by referring to the sign of

\dfrac{dA}{dt}

. (c) Does there exist a time

t>0

at which the spill’s area is momentarily neither increasing nor decreasing? Justify your answer analytically.