Question

6. How many full revolutions does a car tire with a diameter of 25 inches make when the car travels one mile?​

Answer 1

let's convert all units to inches, so hmmm we know the diameter is 25 inches, so that means its radius is half that or 12.5 inches.

let's recall that a mile has 5280 feet, and a feet has 12 inches, so a mile will have (5280)(12) inches total.

so let's reword it.

"how many 360° are there in a circle whose radius is 12.5 inches and has an arc of (5280)(12)?"

`\textit{arc's length}\\\\ s = \cfrac{\theta \pi r}{180} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ r=12.5\\ s=\stackrel{(5280)(12)}{63360} \end{cases}\implies 63360=\cfrac{\theta \pi (12.5)}{180} \\\\\\ (63360)(180)=12.5\pi \theta \implies \cfrac{(63360)(180)}{12.5\pi }=\theta\implies 290420.85^o\approx \theta \\\\\\ \cfrac{290420.85}{\underset{revolution}{360}} ~~ \approx ~~ 806.72\hspace{5em}\stackrel{ \textit{full revolutions} }{\text{\LARGE 806}}`