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A sphere is being manufactured with a radius of 90 cm and a maximum possible error of 0.008 cm for the radius. Find the approximate error and the approximate relative error in the volume of the sphere. Give the relative error as a percent.
The differential is \(\displaystyle dV = 4 \pi r^{2}\). Evaluating at \(\displaystyle r = 90\) and \(\displaystyle dr = 0.008\) gives \(\displaystyle 259.2 \pi\). The relative error is given by \(\displaystyle \frac{dV}{V} = \frac{4 \pi r^{2}}{\frac{4 \pi r^{3}}{3}}\,dr=\frac{3}{r}\,dr\). Evaluating gives \(\displaystyle 0.00027 = 0.027\)%
\begin{question}A sphere is being manufactured with a radius of 90 cm and a maximum possible error of 0.008 cm for the radius. Find the approximate error and the approximate relative error in the volume of the sphere. Give the relative error as a percent.
\soln{9cm}{The differential is $dV = 4 \pi r^{2}$. Evaluating at $r = 90$ and $dr = 0.008$ gives $259.2 \pi$. The relative error is given by $\frac{dV}{V} = \frac{4 \pi r^{2}}{\frac{4 \pi r^{3}}{3}}\,dr=\frac{3}{r}\,dr$. Evaluating gives $0.00027 = 0.027$\%}
\end{question}
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\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>A sphere is being manufactured with a radius of 90 cm and a maximum possible error of 0.008 cm for the radius. Find the approximate error and the approximate relative error in the volume of the sphere. Give the relative error as a percent. </p> </p>
<p> <p>The differential is <img class="equation_image" title=" \displaystyle dV = 4 \pi r^{2} " src="/equation_images/%20%5Cdisplaystyle%20dV%20%3D%204%20%5Cpi%20r%5E%7B2%7D%20" alt="LaTeX: \displaystyle dV = 4 \pi r^{2} " data-equation-content=" \displaystyle dV = 4 \pi r^{2} " /> . Evaluating at <img class="equation_image" title=" \displaystyle r = 90 " src="/equation_images/%20%5Cdisplaystyle%20r%20%3D%2090%20" alt="LaTeX: \displaystyle r = 90 " data-equation-content=" \displaystyle r = 90 " /> and <img class="equation_image" title=" \displaystyle dr = 0.008 " src="/equation_images/%20%5Cdisplaystyle%20dr%20%3D%200.008%20" alt="LaTeX: \displaystyle dr = 0.008 " data-equation-content=" \displaystyle dr = 0.008 " /> gives <img class="equation_image" title=" \displaystyle 259.2 \pi " src="/equation_images/%20%5Cdisplaystyle%20259.2%20%5Cpi%20" alt="LaTeX: \displaystyle 259.2 \pi " data-equation-content=" \displaystyle 259.2 \pi " /> . The relative error is given by <img class="equation_image" title=" \displaystyle \frac{dV}{V} = \frac{4 \pi r^{2}}{\frac{4 \pi r^{3}}{3}}\,dr=\frac{3}{r}\,dr " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7BdV%7D%7BV%7D%20%3D%20%5Cfrac%7B4%20%5Cpi%20r%5E%7B2%7D%7D%7B%5Cfrac%7B4%20%5Cpi%20r%5E%7B3%7D%7D%7B3%7D%7D%5C%2Cdr%3D%5Cfrac%7B3%7D%7Br%7D%5C%2Cdr%20" alt="LaTeX: \displaystyle \frac{dV}{V} = \frac{4 \pi r^{2}}{\frac{4 \pi r^{3}}{3}}\,dr=\frac{3}{r}\,dr " data-equation-content=" \displaystyle \frac{dV}{V} = \frac{4 \pi r^{2}}{\frac{4 \pi r^{3}}{3}}\,dr=\frac{3}{r}\,dr " /> . Evaluating gives <img class="equation_image" title=" \displaystyle 0.00027 = 0.027 " src="/equation_images/%20%5Cdisplaystyle%200.00027%20%3D%200.027%20" alt="LaTeX: \displaystyle 0.00027 = 0.027 " data-equation-content=" \displaystyle 0.00027 = 0.027 " /> %</p> </p>