Question

an oil tanker is leaking oil in the shape of a circle. the function ( r(t)=6sqrt{t} ) determines the radius ( r ) (in feet) of the circle as a function of time ( t ) (in hours). the function ( a(r)=pi r^{2} ) determines the area of a cirlce as a function of the radius.
a. define a new function ( s ) that will determine the surface area of the oil spill as a function of time. put function notation in the first answer box and an appropriate algebraic expression in the second answer box.
(make sure your algebraic expression is simplified.)

b. find the surface area of the oil leak after 26 hours. round your answer to two decimal places.

c. given ( s(30)=3392.92 ), interpret the equation in the context of the problem.

Explanation:

Step1: Find the composition of functions

We know that \(S(t)=A(r(t))\). Since \(r(t) = 6\sqrt{t}\) and \(A(r)=\pi r^{2}\), substitute \(r = 6\sqrt{t}\) into \(A(r)\).

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Step2: Simplify the expression

Using the rule \((ab)^{n}=a^{n}b^{n}\) and \((\sqrt{t})^{2}=t\), we have \((6\sqrt{t})^{2}=6^{2}\times(\sqrt{t})^{2}=36t\). So \(S(t)=\pi\times36t = 36\pi t\)

Step3: Calculate the surface - area at \(t = 26\)

Substitute \(t = 26\) into \(S(t)\). \(S(26)=36\pi\times26\). Since \(\pi\approx3.14159\), then \(S(26)=36\times3.14159\times26\)

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Step4: Interpret \(S(30)=3392.92\)

In the context of the problem, the function \(S(t)\) gives the surface - area of the oil spill as a function of time \(t\). When \(t = 30\) (30 hours have passed), \(S(30)\) represents the surface - area of the oil spill. So the surface - area of the oil spill after 30 hours is 3392.92 square feet.

Answer:

A. \(S(t)=36\pi t\)
B. \(2940.53\)
C. The surface - area of the oil spill after 30 hours is 3392.92 square feet.