Q. 68 - Precalculus Enhanced with Graphing Utilities

Question

The spread of oil leaking from a tanker is in the shape of a circle. If the radius $r$ (in feet) of the spread after $t$ hours is $r\left(t\right)=200\sqrt{t}$, find the area $A$ of the oil slick as a function of the time $t$.

Step-by-Step Solution

Answer

The Area, $\mathrm{A}\left(\mathrm{t}\right)=40,000\mathrm{\pi t}.$

Step 1. Given Information

Given that $r\left(t\right)=200\sqrt{t}.$

Step 2. Solution

We know that the area of circle is $A\left(r\right)={\mathrm{\pi r}}^2.$

We have to find area $A$ as a function of $t.$

Here, $r\left(t\right)=200\sqrt{t}.$

So,

$$\begin{gathered} A\circ r\left(t\right)=A\left(r\right(t\left)\right) \\ A\circ r\left(t\right)=\mathrm{\pi }{\left\{\mathrm{r}\right(\mathrm{t}\left)\right\}}^2\mathrm{A}\circ \mathrm{r}\left(\mathrm{t}\right)=\mathrm{\pi }{\left\{200\sqrt{\mathrm{t}}\right\}}^2\mathrm{A}\circ \mathrm{r}\left(\mathrm{t}\right)=\mathrm{\pi }\left\{40000\mathrm{t}\right\} \\ \mathrm{A}\circ \mathrm{r}\left(\mathrm{t}\right)=40,000\mathrm{\pi t} \\ \mathrm{A}\left(\mathrm{t}\right)=40,000\mathrm{\pi t}. \end{gathered}$$

Step 3. Final answer

Hence, $A$ as a function of $t$ is $\mathrm{A}\left(\mathrm{t}\right)=40,000\mathrm{\pi t}.$