Tangent Planes and Linear Approximations — Question 9

Question 9 – Solution

Strategy. Evaluate the tangent-plane increment $\Delta V\approx V_r\Delta r+V_h\Delta h$ at the nominal dimensions.

Step 1: Volume change $$V_r=2\pi rh,\qquad V_h=\pi r^2.$$ At $(10,20)$, $$dV=(400\pi)(0.03)+(100\pi)(-0.08) =12\pi-8\pi.$$ Thus $$\boxed{\Delta V\approx 4\pi\text{ cm}^3\approx 12.566\text{ cm}^3}.$$

Step 2: Relative change Dividing the differential by $V=\pi r^2h$ gives $$\frac{dV}{V}=2\frac{dr}{r}+\frac{dh}{h} =2\frac{0.03}{10}-\frac{0.08}{20}=0.002.$$ The estimated percentage increase is .

Step 3: Exact comparison $$\Delta V_{\mathrm{exact}} =\pi(10.03)^2(19.92)-2000\pi =\boxed{3.969928\pi\text{ cm}^3}.$$ The linear estimate exceeds the exact change by $$(4-3.969928)\pi=\boxed{0.030072\pi\text{ cm}^3\approx 0.09447\text{ cm}^3}.$$