Question

(1 point) If I drive 16.5 laps around a circular race track and during that drive I go 37 mi, what is the radius of the track

Answer 1

we know that in 16.5 laps we get 37 miles, hmmm well the circumference of that circle is just 1 lap, how many miles is there in 1 lap?

$16.\underline{5}\implies \cfrac{165}{1\underline{0}}\implies \cfrac{33}{2} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{ccll} laps&miles\\ \cline{1-2} \frac{33}{2}&37\\[1em] 1&x \end{array}\implies \cfrac{~~\frac{33}{2} ~~}{1}=\cfrac{37}{x}\implies \cfrac{33}{2}=\cfrac{37}{x} \\\\\\ 33x=74\implies x=\cfrac{74}{33}\impliedby \stackrel{\textit{one lap}~\hfill }{\textit{circumference of the circle}} \\\\[-0.35em] ~\dotfill$

$\textit{Circumference of a circle}\\\\ C=2\pi r~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ C=\frac{74}{33} \end{cases}\implies \cfrac{74}{33}=2\pi r\implies \cfrac{74}{33(2\pi )}=r \\\\\\ \stackrel{miles}{0.35689}\approx r\implies \stackrel{\textit{about 1884 ft and 5 inches}}{1884.395\approx r}$