1. suppose that an offshore oil rig is leaking and that the oil forms a circular region whose radius r…

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.

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.

Answer

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.