Question

a flower garden is shaped like a circle. its radius is 13 yd. a ring - shaped path goes around the garden. its outer edge is a circle with radius 17 yd. the gardener is going to cover the path with sand. if one bag of sand can cover 6 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.)

Explanation:

Step1: Calculate area of outer circle

The formula for the area of a circle is \( A = \pi r^2 \). For the outer circle with radius \( r = 17 \) yd, the area is \( A_{outer} = 3.14\times17^2 \).
\( 17^2 = 289 \), so \( A_{outer} = 3.14\times289 = 907.46 \) square yards.

Step2: Calculate area of inner circle

For the inner circle with radius \( r = 13 \) yd, the area is \( A_{inner} = 3.14\times13^2 \).
\( 13^2 = 169 \), so \( A_{inner} = 3.14\times169 = 530.66 \) square yards.

Step3: Calculate area of the path

The area of the ring - shaped path (annulus) is the area of the outer circle minus the area of the inner circle. So \( A_{path}=A_{outer}-A_{inner} \).
\( A_{path}=907.46 - 530.66=376.8 \) square yards.

Step4: Calculate number of sand bags

One bag of sand covers \( 6 \) square yards. To find the number of bags, we divide the area of the path by the area covered by one bag. Let \( n \) be the number of bags. Then \( n=\frac{A_{path}}{6} \).
\( n = \frac{376.8}{6}=62.8 \). Since the number of bags must be a whole number, we round up to the next whole number (because we can't buy a fraction of a bag and we need to cover the entire path), so \( n = 63 \).

Answer:

63