Chapter 11.6, Problem 30WE

To determine

To determine the length of the belt.

Answer to Problem 30WE

Length of belt is around 184.41cm.

Explanation of Solution

Given information :

  

First we shall find the length of PR using Pythagoras theorem.

We get,

  $\begin{array}{l}P{Q}^2=Q{R}^2+P{R}^2\\ {40}^2={20}^2+P{R}^2\\ 1600-400=P{R}^2\\ PR=\sqrt{1200}=34.64\end{array}$

Now we shall find the angle SQP.

  $\begin{array}{l}\sin \angle SQP=\frac{PR}{PQ}\\ \angle SPQ={60}^{\circ }\end{array}$

Hence the angle TPU is ${120}^{\circ }.$

Length of belt is $2PR+\frac{{120}^{\circ }}{{360}^{\circ }}\left(2\pi \left(5\right)\right)+\frac{{240}^{\circ }}{{360}^{\circ }}\left(2\pi \left(25\right)\right)$

Solving this we get,

  $L=184.41$ cm.