Question

a flower garden is shaped like a circle. its diameter is 30 yd. a ring - shaped path goes around the garden. the width of the path is 5 yd.
the gardener is going to cover the path with sand. if one bag of sand can cover 8 yd², how many bags of sand does the gardener need? note that sand comes only by the bag, so the number of bags must be a whole number. (use the value 3.14 for π.)
bags of sand

Explanation:

Step1: Calculate the radius of the garden

The diameter of the garden is \(d = 30\) yd. The radius of the garden \(r=\frac{d}{2}=\frac{30}{2}=15\) yd.

Step2: Calculate the radius of the garden - path combination

The width of the path is \(5\) yd. So the radius of the outer - circle \(R=r + 5=15 + 5=20\) yd.

Step3: Calculate the area of the 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 \(R^{2}-r^{2}=(20 + 15)(20 - 15)=(35)(5)=175\).
Substituting \(\pi = 3.14\), the area of the path \(A=3.14\times175 = 549.5\) \(yd^{2}\).

Step4: Calculate the number of bags

If one bag of sand covers \(8\) \(yd^{2}\), then the number of bags \(n=\frac{A}{8}\).
\(n=\frac{549.5}{8}=68.6875\).

Since the number of bags must be a whole number, we need to round up.

Answer:

\(69\)