Chapter 4.9, Problem 13P

(a)

To determine

To calculate: Total distance required to run around the track.

Answer to Problem 13P

Around the track,

  $\text{Total}\text{distance}=2\pi r+\pi +8r$

Explanation of Solution

Given information:

Distance to stay from inner edge of the track is 0.5 m.

Total distance required to run includes distance around left and right semicircle as well as straight line distance on the bottom and top of the track.

Since the distance to run from the inner edge is 0.5 m, the left and right semicircles have the same radius of $r+0.5\text{m}$ .

Thereby,

Two semicircles can be combined to make a full circle.

Thus,

The distance around the semicircles:

  $2\pi \left(r+0.5\text{m}\right)=2\pi r+\pi$

Now,

The width of soccer field is 2r .

And

The length of soccer field is twice the width.

Then

The length of soccer field is 4r .

Thereby,

The straight lines on top and bottom of the track are both 4r .

Thus,

The distance around these parts of the track is 8r .

Therefore,

Around the track,

  $\text{Total}\text{distance}=\text{Distance}\text{around}\text{semicircles}+\text{Straight}\text{line}\text{distances}=2\pi r+\pi +8r$

(b)

To determine

To calculate: Farther distance run by the friend.

Answer to Problem 13P

Friend runs $6\pi$ farther.

Explanation of Solution

Given information:

Friend stays 0.5 m from the outer edge of the track.

We know that

Width of the track is 4 m.

If friend runs 0.5 m from the outer edge, he is running 3.5 m from the inner edge.

Then

Friend is running on a semicircle with radius of $r+3.5\text{m}$ .

Thus,

The distance around the semicircles that friend runs:

  $2\pi \left(r+3.5\text{m}\right)=2\pi r+7\pi$

Since the distance from the inner edge doesn’t affect the length, the straight line distances on top and bottom of the track is same for the friend.

Thus,

The straight line distance for the friend is also 8r .

Therefore,

For friend,

  $\text{Total}\text{distance}=\text{Distance}\text{around}\text{semicircles}+\text{Straight}\text{line}\text{distances}=2\pi r+7\pi +8r$

Now,

To know the farther distance that your friend runs than you do, subtract your distance from the distance your friend runs:

  $\begin{array}{l}\text{Farther}\text{distance}=\text{Friend's}\text{distance}-\text{your}\text{distance}\\ \text{=}\left(2\pi r+7\pi +8r\right)-\left(2\pi r+\pi +8r\right)\\ =2\pi r+7\pi +8r-2\pi r-\pi -8r\\ =6\pi \end{array}$