Chapter 4 - Applications of the Derivative - Chapter Review Exercises - Page 221: 9

$$L(r) =36\pi(r-2)$$

Given $$A(r)=\frac{4}{3} \pi r^{3}, \quad a=3$$ Since \begin{align*} A^{\prime}(r)&=4\pi r^2 \\ A^{\prime}(3)&= 36\pi \end{align*} Then the linear approximation is given by \begin{align*} L(r)&=A^{\prime}(a)(r-a)+A(a)\\ &= 36\pi(r-3)+ 36\pi\\ &=36\pi(r-2) \end{align*}