Questions: A fence has to be erected around a rectangular 11 ft by 31 ft swimming pool, 3 feet away from a 3 ft wide sidewalk that surrounds the pool. The fence has two gates, each 3 feet wide. How long is the fence? (Include units in your answer. More information.)

Solution Steps

To determine the length of the fence, we need to calculate the perimeter of the area that includes the swimming pool, the surrounding sidewalk, and the additional 3 feet beyond the sidewalk where the fence will be placed. The fence will have two gates, each 3 feet wide, so we need to subtract the total width of the gates from the calculated perimeter.

1. Calculate the total length and width of the area enclosed by the fence by adding the dimensions of the pool, the sidewalk, and the additional 3 feet on each side.
2. Compute the perimeter of this larger rectangle.
3. Subtract the total width of the gates from the perimeter to get the total length of the fence.

To find the total dimensions of the area enclosed by the fence, we need to account for the swimming pool, the surrounding sidewalk, and the additional 3 feet beyond the sidewalk. The swimming pool is \(31 \, \text{ft}\) long and \(11 \, \text{ft}\) wide. The sidewalk is \(3 \, \text{ft}\) wide, and the fence is placed an additional \(3 \, \text{ft}\) from the sidewalk.

The total length of the area is: \[ \text{Total Length} = 31 + 2 \times (3 + 3) = 43 \, \text{ft} \]

The total width of the area is: \[ \text{Total Width} = 11 + 2 \times (3 + 3) = 23 \, \text{ft} \]

The perimeter of the rectangle that includes the pool, sidewalk, and additional space for the fence is calculated as follows: \[ \text{Perimeter} = 2 \times (43 + 23) = 132 \, \text{ft} \]

The fence includes two gates, each \(3 \, \text{ft}\) wide. Therefore, the total width of the gates is: \[ \text{Total Gate Width} = 2 \times 3 = 6 \, \text{ft} \]

Subtract the total gate width from the perimeter to find the length of the fence: \[ \text{Fence Length} = 132 - 6 = 126 \, \text{ft} \]

Final Answer