Chapter 4.1, Problem 37E

Solution

To identify: The width of the track.

The width is $5.41\text{inches}$

Given information A circular disc of the radius $\text{11}\text{in}$ and $\theta =10$ .

It takes ten identical pieces to form a circular track for a pair of toy racing cars. If the inside arc of each piece is 3.4 in. shorter than the outside arc,

Explanation:

Consider the given information

As the circular track is divided into ten identical pieces.

Consider one such piece:

  $s=\frac{360°}{10}$

Thus, central angle of one piece is $s=\frac{360°}{10}$ as the whole circular track has $360°$ .

Suppose the inner arc length to be $s_1$ and the outer arc length to be $s_2$ .

Thus, $s_2-s_1=3\cdot 4$ inches is given.

Recall that $s=\frac{\pi r\theta }{180°}$ where $r$ is the radius of the circle and $\theta$ is the central angle.

Observe that the central angle for both the inner and outer arc is the same.

Hence, ${\text{s}}_2-s_1=\frac{\left(r_2-r_1\right)\pi \theta }{180}$ .

Therefore,

  $\begin{array}{c}{r}_2-r_1=\frac{\left(s_2-s_1\right)\times 180}{\pi \times \theta }\\ =\frac{3.4\times 180°}{3.14\times 36°}\\ =5.41\end{array}$

As difference in radius of inner and outer arc correspond to the width of the track.

Hence the width of the circle track is $5.41\text{inches}$ .

So, the width is $5.41\text{inches}$ .