## Calculus with Applications (10th Edition)

a. $A[r(t)]=\pi t^{4}$ represents the area of the oil slick as a function of time $t$ b. $D_{t}A[r(t)]=4\pi t^{3}$ represents the rate of change of the spill area at time t. At 100 minutes the area of the spill is changing at the rate of $4,000,000\pi \ \ \mathrm{f}\mathrm{t}^{2}/ \min$
a. $A[r(t)]$ represents the area of the oil slick as a function of time $t$ (t=the time in minutes passed after the beginning of the leak.) $A(r)=\pi r^{2}$ In the definition of A(r), replace r with r(t)\begin{align*} A[r(t)]&=\pi[r(t)]^{2} \quad & \\ &=\pi[t^{2}]^{2} \quad & \\ & =\pi t^{4} \end{align*} This function b. $D_{t}A[r(t)]$ represents the rate of change of the spill area at time t. $D_{t}A[r(t)]=D_{t}[\pi t^{4}]=4\pi t^{3}$ For t=100, $D_{t}A[r(100)]=4\pi(100)^{3}=4,000,000\pi$ At 100 minutes the area of the spill is changing at the rate of $4,000,000\pi \mathrm{f}\mathrm{t}^{2}/$min