## University Calculus: Early Transcendentals (3rd Edition)

We need to keep the temperature in the range $[65^oF, 75^oF]$, or within $5^oF$ of $t_0=70^oF$, so that the width tolerance be maintained.
$$y=10+(t-70)\times10^{-4}$$ To find how close to $t_0=70^oF$ so that the width tolerance $\epsilon$ be not violated from the ideal $10cm$, basically is to find an interval $[a, b]$ of $t$ values such that for all $t\in[a,b]$: $$|y-10|\le\epsilon$$ $$|10+(t-70)\times10^{-4}|\le\epsilon$$ $$|(t-70)\times10^{-4}|\le\epsilon$$ $$-\epsilon\le(t-70)\times10^{-4}\le\epsilon$$ $$\frac{-\epsilon}{10^{-4}}\le(t-70)\le\frac{\epsilon}{10^{-4}}$$ $$-10^4\epsilon\le(t-70)\le10^4\epsilon$$ $$70-10^4\epsilon\le t\le70+10^4\epsilon$$ The width tolerance $\epsilon$ here is $0.0005cm$. Therefore, $$70-10^4\times0.0005\le t\le70+10^4\times0.0005$$ $$70-5\le t\le70+5$$ $$65\le t\le75$$ This means we need to keep the temperature in the range $[65^oF, 75^oF]$, or within $5^oF$ of $t_0=70^oF$, so that the width tolerance be maintained.