A circular pool has a diameter of 24 yards. A ring-shaped path with a width of 6 yards surrounds the pool. We need to determine how many gallons of coating are required to cover the path, given that one gallon covers 8 square yards. We are told to use 3.14 for $\pi$. The number of gallons must be a whole number.

1. Problem Description

A circular pool has a diameter of 24 yards. A ring-shaped path with a width of 6 yards surrounds the pool. We need to determine how many gallons of coating are required to cover the path, given that one gallon covers 8 square yards. We are told to use 3.14 for $\pi$. The number of gallons must be a whole number.

2. Solution Steps

First, we need to find the radius of the pool and the outer radius of the path.

The diameter of the pool is 24 yards, so the radius of the pool is $r_1 = \frac{24}{2} = 12$ yards.

The path has a width of 6 yards, so the outer radius is $r_2 = r_1 + 6 = 12 + 6 = 18$ yards.

Next, we need to find the area of the path. The area of the path is the difference between the area of the outer circle and the area of the inner circle (the pool).

Area of the outer circle:

$A_2 = \pi r_2^2 = \pi (18)^2 = \pi (324)$

Area of the inner circle (pool):

$A_1 = \pi r_1^2 = \pi (12)^2 = \pi (144)$

Area of the path:

$A = A_2 - A_1 = \pi(324) - \pi(144) = \pi (324 - 144) = \pi (180)$

We are given that $\pi = 3.14$, so:

$A = 3.14 (180) = 565.2$ square yards.

Finally, we need to determine the number of gallons of coating required.

One gallon covers 8 square yards, so we divide the area of the path by 8 to find the number of gallons needed:

Number of gallons $= \frac{565.2}{8} = 70.65$

Since the coating comes only by the gallon, we must round up to the nearest whole number to ensure we have enough coating.

Number of gallons required $= 71$

3. Final Answer

71 gallons of coating