An automobile tire has a diameter of 36 inches. How many revolutions will the wheel make as the...

Question:

An automobile tire has a diameter of 36 inches. How many revolutions will the wheel make as the automobile travel 1 mile?

Circumference

The circumference is the linear length obtained by converting a circle into a line. The circumference depends on the radius or diameter of the circle. The general expression for the circumference is {eq}C = 2\pi r = \pi d {/eq}.

Answer and Explanation:

Given Data

• The diameter of the tire is: {eq}r = 36\;{\rm{in}} {/eq}

• The traveled distance is: {eq}s = 1\;{\rm{mile}} {/eq}

The radius of the tire is,

{eq}r = \dfrac{d}{2} {/eq}

The circumference of the tire is,

{eq}\begin{align*} C &= 2\pi r\\ &= 2\pi \left( {\dfrac{d}{2}} \right)\\ &= \pi d \end{align*} {/eq}

The expression for the number of revolutions is,

{eq}N = \dfrac{s}{C} {/eq}

Substitute the known values,

{eq}\begin{align*} N &= \dfrac{{1\;{\rm{mile}}}}{{\pi d}}\\ &= \dfrac{{1\;{\rm{mile}}}}{{\pi \left( {36\;{\rm{in}}} \right)}}\left( {\dfrac{{63360\;{\rm{in}}}}{{1\;{\rm{mile}}}}} \right)\\ &= 560.225\;{\rm{revolutions}}\\ &\approx 560\;{\rm{revolutions}} \end{align*} {/eq}

Thus, the number of revolutions is {eq}560\;{\rm{revolutions}} {/eq}.