At a local park, there is a large circular fountain feature that is 50 feet across,

surrounded by a path that is 8ft wide. A father and son plan to have a “friendly”
race around the path. To be fair, the father agrees to run around the outside
circle of the path, and the son will run on the inside next to the fountain.
a) How far will the son have to run to make one circuit around the fountain?
b) Because the father is running around the outside of the path, he will
have to run farther. Determine how much farther he will have to run to
make one circuit of the fountain.

See previous post.

To determine how far the son will have to run to make one circuit around the fountain, we need to calculate the length of the path that is on the inside next to the fountain.

a) The son will have to run on the inside of the path, so the length of the path he will run will be the circumference of the circle with a radius of 50ft minus the circumference of the circle with a radius of 50ft minus 8ft.

To calculate the circumference of a circle, we use the formula C = 2πr, where C is the circumference and r is the radius.

The circumference of the larger circle is C1 = 2π(50ft) = 100π ft.

The circumference of the smaller circle is C2 = 2π(50ft - 8ft) = 84π ft.

Therefore, the length of the path the son will have to run is C1 - C2 = 100π ft - 84π ft = 16π ft.

So, the son will have to run approximately 16π ft to make one circuit around the fountain.

b) To determine how much farther the father will have to run to make one circuit around the fountain, we need to calculate the length of the path he will run on the outside.

Since the father is running on the outside circle of the path, he will run the full circumference of the larger circle.

Using the same formula for the circumference of a circle, the length of the path the father will have to run is C1 = 100π ft.

Therefore, the father will have to run approximately 100π ft to make one circuit around the fountain.

To determine how much farther the father will have to run compared to the son, we subtract the length of the son's path from the father's path.

100π ft - 16π ft = 84π ft.

So, the father will have to run approximately 84π ft farther than the son to make one circuit around the fountain.