Question

you try 4.7. a tire on a 2014 dodge durango suv has an outside diameter of 36.32 in. through what angle (in radian) does the tire turn while traveling 1 mi?

Explanation:

Step1: Convert units

First, convert the distance traveled from miles to inches. Since \(1\) mile \( = 5280\) feet and \(1\) foot \(= 12\) inches, then \(s = 1\times5280\times12=63360\) inches. The radius \(r\) of the tire is \(r=\frac{d}{2}\), where \(d = 36.32\) inches, so \(r=\frac{36.32}{2}=18.16\) inches.

Step2: Use the arc - length formula

The arc - length formula is \(s = r\theta\), where \(s\) is the arc length (distance traveled), \(r\) is the radius, and \(\theta\) is the angle in radians. We need to solve for \(\theta\). Rearranging the formula gives \(\theta=\frac{s}{r}\).
Substitute \(s = 63360\) inches and \(r = 18.16\) inches into the formula: \(\theta=\frac{63360}{18.16}\).
Calculate \(\frac{63360}{18.16}=\frac{6336000}{1816}=\frac{6336000\div16}{1816\div16}=\frac{396000}{113.5}\approx3489.87\) radians.

Answer:

The tire turns approximately \(3490\) radians.