Chapter 4.1, Problem 57E

Solution

To identify: The statement is true or false. If horse A is twice as far as horse B from the center or a merry-go-round, then horse A travel twice as fast as horse B.

This statement is true.

Given information:

It is given that the horse A is twice as far as horse B from the center or a merry-go-round, then horse A travel twice as fast as horse B.

Explanation:

Consider the given information.

Answer to above is true, because

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

Observe that from the statement it is clear that the central angle for the two horses A and B is same and the radius for horse A is twice the radius for horse B.

That is if radius of horse B is r then radius of horse A is 2r .

Therefore, from the above formula for lengths s of the intercepted arc:

  $\begin{array}{c}{s}_A=\frac{\pi 2r\theta }{180}\\ s_A=2\left(\frac{\pi r\theta }{180}\right)\\ =2s_B\end{array}$

Therefore, horse A travel twice as fast as horse B in one sector is true.