Question

at the park there is a pool shaped like a circle with diameter 18 yd. a ring - shaped path goes around the pool. its width is 5 yd. we are going to give a new layer of coating to the path. if one gallon of coating can cover 4 yd², how many gallons of coating do we need? note that coating comes only by the gallon, so the number of gallons must be a whole number. (use the value 3.14 for π.)

Explanation:

Step1: Calculate the radius of the inner circle

The diameter of the inner circle (pool) is \(d = 18\) yd. The radius of the inner circle \(r=\frac{d}{2}=\frac{18}{2}=9\) yd.

Step2: Calculate the radius of the outer circle

The width of the path is \(5\) yd. So the radius of the outer circle \(R=r + 5=9 + 5=14\) yd.

Step3: Calculate the area of the ring - shaped path

The area of a circle is \(A=\pi R^{2}-\pi r^{2}=\pi(R^{2}-r^{2})\). Using the formula \(a^{2}-b^{2}=(a + b)(a - b)\), we have \(A = 3.14\times(14^{2}-9^{2})=3.14\times(14 + 9)\times(14 - 9)\).
First, \(14+9 = 23\) and \(14 - 9=5\). Then \(A=3.14\times23\times5=3.14\times115 = 351.1\) \(yd^{2}\).

Step4: Calculate the number of gallons of coating

Since one gallon of coating can cover \(4\) \(yd^{2}\), the number of gallons \(n=\frac{A}{4}=\frac{351.1}{4}=87.775\). Since the number of gallons must be a whole number, we round up.

Answer:

\(88\)