Question

a sprinkler is designed to water a circular area that has a radius of 7 feet. the sprinkler is located at (0,0) on a lawn. a flowerbed is planted 3 feet east and 6 feet south of the sprinkler. determine if the distance from the flowerbed to the sprinkler places the flowerbed on or within the circular area that this sprinkler can water. no. the flowerbed is 18 feet away from the sprinkler, which is more than the 7 - foot radius. yes. the flowerbed is exactly 7 feet away from the sprinkler. no. the flowerbed is 9 feet away from the sprinkler, which is more than the 7 - foot radius. yes. the flowerbed is \\( \sqrt{45} \\) feet away from the sprinkler, which is less than the 7 - foot radius.

Explanation:

Step1: Find the coordinates of the flowerbed

Since the sprinkler is at \((0,0)\), and the flowerbed is \(3\) feet east (positive \(x\)-direction) and \(6\) feet south (negative \(y\)-direction), the coordinates of the flowerbed are \((3,-6)\).

Step2: Use the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(3,-6)\). So \(d=\sqrt{(3 - 0)^2+(-6 - 0)^2}=\sqrt{9 + 36}=\sqrt{45}\approx6.71\) feet.

Step3: Compare the distance with the radius

The radius of the circular area is \(r = 7\) feet. Since \(\sqrt{45}\approx6.71<7\).

Answer:

Yes. The flowerbed is \(\sqrt{45}\) feet away from the sprinkler, which is less than the 7 - foot radius.