Question

Rob just installed 8 water sprinklers in his front yard. Each one rotates 360 degrees while spraying out water reaching out 13 feet from the sprinkler.

a) How much area does each sprinkler cover?
b) How much area is covered by all 8 sprinklers combined?​

Answer 1

We can use the following formulas to solve the problem:

The area of a circle is given by the formula A = πr^2, where r is the radius of the circle.

The area of a sector of a circle is given by the formula A = (θ/360)πr^2, where θ is the central angle of the sector in degrees.

a) Each sprinkler rotates 360 degrees, so it covers a full circle with a radius of 13 feet. Therefore, the area covered by each sprinkler is:

A = πr^2 = π(13 ft)^2 = 169π sq ft ≈ 530 sq ft (rounded to the nearest whole number)

b) Since there are 8 sprinklers, the total area covered by all the sprinklers is 8 times the area covered by one sprinkler:

Total area = 8 x 530 sq ft = 4,240 sq ft

Therefore, all 8 sprinklers combined cover an area of 4,240 square feet.

Answer 2

Answer: 4,240 square feet.

Explanation:

Step 1

a) Each sprinkler covers an area of approximately 530 square feet (pi times the square of the radius, where radius is half of the distance reached by the sprinkler).

b) All 8 sprinklers combined cover an area of approximately 4,240 square feet (530 square feet times 8).

Step 2

To find the area covered by each sprinkler,

we need to calculate the area of the circle formed by the water spray. The radius of the circle is the distance reached by the sprinkler, which is 13 feet. So, the area covered by each sprinkler is π(13)^2 square feet, which is approximately equal to 530.93 square feet.

Since there are 8 sprinklers, the total area covered by all of them is simply 8 times the area covered by each sprinkler. Therefore, the total area covered by all 8 sprinklers combined is 8 x 530.93 square feet, which is equal to 4,247.44 square feet.

Step 3

To answer part A of the problem, we need to first calculate the area covered by each sprinkler. Since each sprinkler covers a circular area with a radius of 13 feet, we can use the formula for the area of a circle (πr^2) to find the area covered by each sprinkler, which is approximately 530.93 square feet. For part B, we simply need to multiply the area covered by each sprinkler by the number of sprinklers, giving us a total coverage area of about 4,247.43 square feet for all 8 sprinklers.