Question

A rectangular lawn has an area of 126 square meters. Surrounding the entire lawn is a flower border that is 4 meters wide. The area of the flower border alone (not including the lawn) is 264 square meters. A sprinkler is installed in the middle of the lawn that sprays water in a circle. Rounded to the nearest 10th of a meter, what is the minimum spraying radius necessary in order for the sprinkler to water both the lawn AND the entire flower border?

[10 points]

Answer 1

To find the minimum spraying radius necessary for the sprinkler to water both the lawn and the entire flower border, we calculate the combined area of the lawn and the flower border and use it to find the radius of a circle.

Step-by-step explanation:

To find the minimum spraying radius needed to water both the lawn and the entire flower border, we need to consider the combined area of the lawn and the flower border. The area of the lawn is given as 126 square meters, while the area of the flower border is given as 264 square meters. Therefore, the combined area is 126 square meters + 264 square meters = 390 square meters.

Since the sprinkler waters in a circle, we can calculate the minimum spraying radius by finding the radius of a circle with an area of 390 square meters. Using the formula for the area of a circle, A = πr^2, we can rearrange the equation to solve for the radius:

r = √(A/π) = √(390/π) ≈ 11.15 meters.

Therefore, the minimum spraying radius necessary for the sprinkler to water both the lawn and the flower border is approximately 11.15 meters.

Answer 2

To solve this problem, we can first use the area of the flower border to calculate the perimeter of the flower border. Since the flower border is 4 meters wide and has an area of 264 square meters, we know that it has a perimeter of 264 / 4 = 66 meters.

Next, we can use the perimeter of the flower border to calculate the dimensions of the lawn. Since the total perimeter of the lawn and flower border is 66 meters and the width of the flower border is 4 meters, we know that the length and width of the lawn must be 31 meters and 5 meters, respectively.

Finally, we can use the dimensions of the lawn to calculate the radius of the sprinkler. Since the lawn is a rectangle with a length of 31 meters and a width of 5 meters, we know that the center of the lawn is located 15.5 meters from each of the longer sides and 2.5 meters from each of the shorter sides. Therefore, the minimum radius of the sprinkler must be 15.5 meters in order to reach the entire flower border.

Rounded to the nearest tenth of a meter, the minimum spraying radius necessary is 15.5 meters.