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.
a. C = pi*D = 3.14 * 50Ft = 157 Ft. =
Circumference = Distance the son must
run.
b. C = 3.14 * 66Ft. = 207 Ft. = Dist.
father must run.
207 - 157 = 50 Ft. Farther than the son.
a) To find out how far the son will have to run to make one circuit around the fountain, we need to calculate the circumference of the circle formed by the inside of the path.
The diameter of the circle formed by the inside of the path is equal to the diameter of the fountain minus twice the width of the path. So, the diameter of the inside circle is 50ft - 2(8ft) = 50ft - 16ft = 34ft.
The circumference of a circle is calculated using the formula C = 2πr, where C is the circumference and r is the radius.
Since the diameter of the inside circle is 34ft, the radius is half of that, which is 34ft / 2 = 17ft.
Plugging in the values into the formula, we get C = 2π(17ft) = 34πft.
Therefore, the son will have to run approximately 34πft to make one circuit around the fountain.
b) To determine how much farther the father will have to run to make one circuit of the fountain, we need to calculate the circumference of the circle formed by the outside of the path.
The diameter of the circle formed by the outside of the path is equal to the diameter of the fountain plus twice the width of the path. So, the diameter of the outside circle is 50ft + 2(8ft) = 50ft + 16ft = 66ft.
Using the same formula C = 2πr, where the radius is half of the diameter, we find the radius of the outside circle is 66ft / 2 = 33ft.
Substituting the values into the formula, we get C = 2π(33ft) = 66πft.
Therefore, the father will have to run approximately 66πft to make one circuit of the fountain.
To find out how much farther the father will have to run, we subtract the distance the son has to run from the distance the father has to run. Thus, the difference is:
66πft - 34πft = 32πft.
Therefore, the father will have to run approximately 32πft farther to make one circuit of the fountain compared to the son.