Chapter 3: Derivatives - Section 3.9 - Linearization and Differentials - Exercises 3.9 - Page 175: 49

$3\%$

Step 1. Using the volume formula $V=4\pi r^3/3$, we have $dV=4\pi r^2 dr$ and $\frac{dV}{V}=\frac{4\pi r^2 dr}{4\pi r^3/3}=\frac{3dr}{r}$ Step 2. Given $D=100\pm1cm$ for the diameter, we have $2r=100\pm1$ and $r=50\pm0.5cm$, we have $dr=0.5cm$ and $\frac{dr}{r}=\frac{0.5}{50}=1\%$ Step 3. Thus, we have $\frac{dV}{V}=\frac{3dr}{r}=3\%$