\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
A cube is being manufactured with a side length of 97 cm and a maximum possible error of 0.012 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 \(\displaystyle s = 97\) and \(\displaystyle ds = 0.012\) gives \(\displaystyle 338.724\). The relative error is given by \(\displaystyle \frac{dV}{V} = \frac{3 s^{2}}{s^{3}}\,ds=\frac{3}{s}\,ds\). Evaluating gives \(\displaystyle 0.00037 = 0.037\)%
\begin{question}A cube is being manufactured with a side length of 97 cm and a maximum possible error of 0.012 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 $s = 97$ and $ds = 0.012$ gives $338.724$. The relative error is given by $\frac{dV}{V} = \frac{3 s^{2}}{s^{3}}\,ds=\frac{3}{s}\,ds$. Evaluating gives $0.00037 = 0.037$\%}
\end{question}
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}
\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}
\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}
\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue
\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>A cube is being manufactured with a side length of 97 cm and a maximum possible error of 0.012 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 = 97 " src="/equation_images/%20%5Cdisplaystyle%20s%20%3D%2097%20" alt="LaTeX: \displaystyle s = 97 " data-equation-content=" \displaystyle s = 97 " /> and <img class="equation_image" title=" \displaystyle ds = 0.012 " src="/equation_images/%20%5Cdisplaystyle%20ds%20%3D%200.012%20" alt="LaTeX: \displaystyle ds = 0.012 " data-equation-content=" \displaystyle ds = 0.012 " /> gives <img class="equation_image" title=" \displaystyle 338.724 " src="/equation_images/%20%5Cdisplaystyle%20338.724%20" alt="LaTeX: \displaystyle 338.724 " data-equation-content=" \displaystyle 338.724 " /> . 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 0.00037 = 0.037 " src="/equation_images/%20%5Cdisplaystyle%200.00037%20%3D%200.037%20" alt="LaTeX: \displaystyle 0.00037 = 0.037 " data-equation-content=" \displaystyle 0.00037 = 0.037 " /> %</p> </p>