We are tasked with finding how many gallons of coating are needed for a ring-shaped path around the pool. Let's break down the solution:

Step 1: Calculate the area of the ring-shaped path

The area of the ring is the difference between the area of the larger circle (outer radius) and the smaller circle (inner radius).

• Area of a circle is given by the formula $A = \pi r^2$.

  • Outer radius ($r_{outer}$) = 14 yd
  • Inner radius ($r_{inner}$) = 10 yd

  $A_{outer} = \pi (14)^2 = 3.14 \times 196 = 615.44 \, \text{yd}^2$ $A_{inner} = \pi (10)^2 = 3.14 \times 100 = 314 \, \text{yd}^2$

• The area of the ring-shaped path is: $A_{path} = A_{outer} - A_{inner} = 615.44 - 314 = 301.44 \, \text{yd}^2$

Step 2: Determine the number of gallons needed

• Each gallon of coating covers 8 yd².
• To find the number of gallons required: $\text{Gallons needed} = \frac{301.44}{8} = 37.68$

Since the number of gallons must be a whole number, we round up to 38 gallons.

Final Answer:

You need 38 gallons of coating.

Would you like more details on any of these steps?

Here are five related questions you might find useful:

1. How would the calculation change if the inner radius was larger?
2. What if the coating covered only 5 yd² per gallon?
3. How do you calculate the circumference of the inner and outer circles?
4. How would the required coating change if the pool had a rectangular shape?
5. Could you apply this method to calculate areas of other shapes like ellipses?

Tip: Always round up when dealing with materials that need to fully cover a surface!