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A cube is being manufactured with a side length of 96 cm and a maximum possible error of 0.076 cm for the sides. Find the approximate error and the approximate relative error in the volume of the cube. Give the relative error as a percent.
The differential is \(\displaystyle dV = 3 s^{2}\). Evaluating at \(\displaystyle s = 96\) and \(\displaystyle ds = 0.076\) gives \(\displaystyle 2101.248\). The relative error is given by \(\displaystyle \frac{dV}{V} = \frac{3 s^{2}}{s^{3}}\,ds=\frac{3}{s}\,ds\). Evaluating gives \(\displaystyle 0.00238 = 0.238\)%
\begin{question}A cube is being manufactured with a side length of 96 cm and a maximum possible error of 0.076 cm for the sides. Find the approximate error and the approximate relative error in the volume of the cube. Give the relative error as a percent.
\soln{9cm}{The differential is $dV = 3 s^{2}$. Evaluating at $s = 96$ and $ds = 0.076$ gives $2101.248$. The relative error is given by $\frac{dV}{V} = \frac{3 s^{2}}{s^{3}}\,ds=\frac{3}{s}\,ds$. Evaluating gives $0.00238 = 0.238$\%}
\end{question}
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\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>A cube is being manufactured with a side length of 96 cm and a maximum possible error of 0.076 cm for the sides. Find the approximate error and the approximate relative error in the volume of the cube. Give the relative error as a percent. </p> </p>
<p> <p>The differential is <img class="equation_image" title=" \displaystyle dV = 3 s^{2} " src="/equation_images/%20%5Cdisplaystyle%20dV%20%3D%203%20s%5E%7B2%7D%20" alt="LaTeX: \displaystyle dV = 3 s^{2} " data-equation-content=" \displaystyle dV = 3 s^{2} " /> . Evaluating at <img class="equation_image" title=" \displaystyle s = 96 " src="/equation_images/%20%5Cdisplaystyle%20s%20%3D%2096%20" alt="LaTeX: \displaystyle s = 96 " data-equation-content=" \displaystyle s = 96 " /> and <img class="equation_image" title=" \displaystyle ds = 0.076 " src="/equation_images/%20%5Cdisplaystyle%20ds%20%3D%200.076%20" alt="LaTeX: \displaystyle ds = 0.076 " data-equation-content=" \displaystyle ds = 0.076 " /> gives <img class="equation_image" title=" \displaystyle 2101.248 " src="/equation_images/%20%5Cdisplaystyle%202101.248%20" alt="LaTeX: \displaystyle 2101.248 " data-equation-content=" \displaystyle 2101.248 " /> . The relative error is given by <img class="equation_image" title=" \displaystyle \frac{dV}{V} = \frac{3 s^{2}}{s^{3}}\,ds=\frac{3}{s}\,ds " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7BdV%7D%7BV%7D%20%3D%20%5Cfrac%7B3%20s%5E%7B2%7D%7D%7Bs%5E%7B3%7D%7D%5C%2Cds%3D%5Cfrac%7B3%7D%7Bs%7D%5C%2Cds%20" alt="LaTeX: \displaystyle \frac{dV}{V} = \frac{3 s^{2}}{s^{3}}\,ds=\frac{3}{s}\,ds " data-equation-content=" \displaystyle \frac{dV}{V} = \frac{3 s^{2}}{s^{3}}\,ds=\frac{3}{s}\,ds " /> . Evaluating gives <img class="equation_image" title=" \displaystyle 0.00238 = 0.238 " src="/equation_images/%20%5Cdisplaystyle%200.00238%20%3D%200.238%20" alt="LaTeX: \displaystyle 0.00238 = 0.238 " data-equation-content=" \displaystyle 0.00238 = 0.238 " /> %</p> </p>