Question

A circular oil spill continues to increase in size. The radius of the oil spill, in miles, is given by the function r(t) = 0.5 + 2t, where t is the time in hours. The area of the circular region is given by the function A(r) = πr2, where r is the radius of the circle at time t.

Explain how to write a composite function to find the area of the region at time t.

Answer 1

You compose A with r by substituting r(t) into A(r):

A(t) = A(r(t)) = π [r(t)]^2 = π(0.5 + 2t)^2.

You can expand if you like: A(t) = π(4t^2 + 2t + 0.25). (Area in square miles; t≥0.)