Question

1. suppose that an offshore oil rig is leaking and that the oil forms a circular region whose radius r increases over the first 12 hours according to the function r = f(t)=\\(\frac{t}{2t + 4}\\), 0 < t ≤ 12 where r is measured in miles, t is in hours, and t = 0 corresponds to the instant that the leak begins. (a) using function notation, the area of a circle of radius r is given by a(r)=\\(\pi r^{2}\\). find (a ∘ f)(t) and interpret the result.

Explanation:

Step1: Recall the composition of functions

The composition \((A\circ f)(t)=A(f(t))\). We know \(A(r)=\pi r^{2}\) and \(r = f(t)=\frac{t}{2t + 4}\) for \(0\leq t<12\).

Step2: Substitute \(r = f(t)\) into \(A(r)\)

Substitute \(r=\frac{t}{2t + 4}\) into \(A(r)\). So \(A(f(t))=\pi(\frac{t}{2t + 4})^{2}=\frac{\pi t^{2}}{(2t + 4)^{2}}\), for \(0\leq t<12\).

Step3: Interpret the result

The function \((A\circ f)(t)\) gives the area of the circular - shaped oil - spill region in square miles at time \(t\) hours (where \(0\leq t<12\)). It represents how the area of the oil - spill changes over the first 12 hours of the leak.

Answer:

\((A\circ f)(t)=\frac{\pi t^{2}}{(2t + 4)^{2}}\), for \(0\leq t<12\). It represents the area of the circular oil - spill region in square miles at time \(t\) hours during the first 12 hours of the leak.