Chapter 12.2, Problem 32PPS

To determine

Calculate the arc length of the circle made by paddlewheel.

Answer to Problem 32PPS

The length of the arc of the circle is $62.83ft$ .

Explanation of Solution

Given: It is given in the question that the paddlewheel of a river boat has a diameter of $24$ feet. And it rotates with an angle of ${300}^{\circ }$ .

Concept Used:

In this, use the concept of changing degree into radian by multiplying $\frac{\pi }{180}$ and use the formula of arc length $s=r\cdot \theta$ .

Calculation: Here, the diameter is $24$ feet and the radius will be $\frac{24}{2}=12ft$ .

Now, convert the degree into radian,

  ${300}^{\circ }=300\times \frac{\pi }{180}=\frac{30\pi }{18}=\frac{5\pi }{3}$

Now, use the arc length formula,

  $\begin{array}{l}s=r\cdot \theta \\ s=12ft\cdot \frac{5\pi }{3}\\ s=4ft\cdot 5\pi \\ s=20\pi ft\\ s=62.83ft\end{array}$

Conclusion:

The length of the arc is $62.83ft$ .