A row of sprinklers is to be installed parallel to and 4.5 feet away from the back edge of a flower bed. Each sprinkler waters a region with a 6 foot radius. How far apart should the sprinklers be placed to water the entire flower bed with the least possible overlap in coverage?

How would I set up the problem?

To set up this problem, let's break it down step by step:

Step 1: Determine the width of the flower bed.
To do this, you need to know the distance between the back edge of the flower bed and the front edge. Without this information, it is not possible to determine the width accurately.

Step 2: Calculate the coverage of each sprinkler.
Each sprinkler has a watering range with a 6-foot radius. This means that each sprinkler covers a circular area with a diameter of 2 times the radius, which is 12 feet.

Step 3: Determine the distance between each sprinkler.
To minimize overlap in coverage, the distance between neighboring sprinklers should be equal to the diameter of the sprinkler's coverage area. In this case, it is 12 feet.

Step 4: Calculate the minimum number of sprinklers needed.
To cover the entire flower bed without any gaps, you need to determine the number of sprinklers required along the width of the flower bed. Divide the width of the flower bed by the distance between each sprinkler to get the minimum number of sprinklers needed.

However, without knowing the precise width of the flower bed, it is not possible to calculate the number of sprinklers accurately.

So, to solve the problem accurately, you need to know the width of the flower bed and then follow the steps outlined above.

To set up the problem, we can represent the flower bed as a rectangle. Let's assume the width of the flower bed is w feet. The sprinklers will be installed parallel to the back edge of the flower bed, 4.5 feet away.

Since each sprinkler waters a region with a 6 foot radius, it will create a circle with a diameter of 12 feet. We want to place the sprinklers in such a way that they cover the entire flower bed with the least possible overlap in coverage. This means that the circles formed by the sprinklers should cover the entire width of the flower bed without overlapping.

To calculate the distance between two adjacent sprinklers, we need to consider the diameter of the circle created by each sprinkler. Since the sprinklers are placed 4.5 feet away from the back edge of the flower bed, the distance between the centers of two adjacent sprinklers should be the sum of the diameter of the circle and the width of the flower bed:

Distance between sprinklers = Diameter of circle + Width of flower bed

Let's solve the problem step-by-step to find the distance between the sprinklers.