Question

A circular oil slick spreads so that as its radius changes, its area changes. Both the radius r and the area A change with respect to time. If dr/dt is found to be 1.7 m/hr, find dA/dt when r= 39.8 m. Hint: A(r)= πr², and, using the Chain, dA/dt=dA/dr•dr/dt.

Answer 1

Given:

dr/dt = 1.7 m/hr

r = 39.8 m

Let's solve for dA/dt when r is 39.8 m.

Where:

`A(r)=\pi r^2`

Now, let's find the derivative.

We have:

`(dA)/(dt)=2\pi r(dr)/(dt)`

Now, substitute 39.8 for r and 1.7 for dr/dt to solve for dA/dt:

`\begin{gathered} (dA)/(dt)=2\pi\ast39.8\ast1.7 \\ \\ (dA)/(dt)=2\pi\ast67.66 \\ \\ (dA)/(dt)=425.12m^2\text{ /hr} \end{gathered}`

ANSWER:

425.12 m²/hr