Problem 92 - Algebra and Trigonometry

Question

The radius of a wheel rolling on the ground is 80 centimeters. If the wheel rotates through an angle of \(60^{\circ},\) how many centimeters does it move? Express your answer in terms of \(\pi\) and then round to two decimal places.

Step-by-Step Solution

Answer

The wheel moves approximately 83.78 cm

Step 1: Convert angle to radians

The given angle is in degrees, which needs to be converted to radians before using the formula. The conversion formula is \(\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}\). So, \(\theta = 60^{\circ} \times \frac{\pi}{180} = \frac{\pi}{3}\) radians.

Step 2: Use arc length formula

The formula for arc length is \(L = r \times \theta\). Here, \(r = 80\) cm and \(\theta = \frac{\pi}{3}\) radians, obtained from step 1. Substitute these values in the formula to get \(L = 80 \times \frac{\pi}{3} = \frac{80\pi}{3}\) cm.

Step 3: Round to two decimal places

The solution requires rounding the result to two decimal places. Hence, calculate \(\frac{80\pi}{3}\) and round to two decimal places. The final solution is approximately 83.78 cm.