To answer the question, we have to determine the area of the path and notice that the area of the path is equal to the area of the outer circle minus the area of the inner circle.
Recall that the area of a circle can be determined using the following formula:
[tex]A=\pi\text{ r}^2,[/tex]where r is the radius of the circle.
Therefore, the area of the outer circle is:
[tex]A_1=(10yd)^2\pi.[/tex]The area of the inner circle is:
[tex]A_2=(8yd)^2\pi.[/tex]Therefore, the area of the ring is:
[tex]A_1-A_2=36\pi yd^2.[/tex]Now, each gallon can cover 7 yd², therefore, you need:
[tex]\frac{36\pi}{7}[/tex]gallons to cover the entire ring.
Answer: Note: since the number of gallons has to be a whole number but we need to cover the entire path, the number is rounded up to the nearest whole number.
[tex]17\text{ gallons.}[/tex]