Question

your community decides to install a rectangular swimming pool in a park. there is a 48-foot by 81-foot rectangular area available. there must be at least a 3-foot border around the pool. identify the diagram of this situation. diagrams (omitted) showing four coordinate plane graphs with rectangles labeled pool and border based on the constraints, find the perimeter and the area of the largest swimming pool possible.

Explanation:

Step1: Determine pool dimensions

The available area is 48 ft (width) by 81 ft (length). With a 3 - foot border on both sides:

• Width of pool: \(48 - 2\times3=42\) ft (subtract 3 ft from left and right).
• Length of pool: \(81 - 2\times3 = 75\) ft (subtract 3 ft from top and bottom). Wait, but looking at the diagrams, the x - dimension of the available area is 48 (from \(x = 0\) to \(x = 48\)) and y - dimension: let's check the correct border. Wait, in the diagrams, the bottom - left corner of the available area is \((0,0)\) and top - right is \((48,81)\) (the last diagram has \((0,81)\) as top - left of available area). Wait, the available area is 48 ft (width, x - direction) and 81 ft (length, y - direction). For the border, 3 ft on each side (left, right, top, bottom). So:
• Pool width (x - direction): \(48-3 - 3=42\) ft (since from \(x = 3\) to \(x = 45\), \(45 - 3=42\)).
• Pool length (y - direction): \(81-3 - 3 = 75\)? Wait no, in the last diagram, the top of available area is \(y = 81\), and the pool top is \(y = 78\), so \(81 - 3=78\) (top border 3 ft), and bottom is \(y = 3\), so \(3 - 0 = 3\) (bottom border 3 ft), so length of pool is \(78 - 3=75\)? Wait no, \(78-3 = 75\)? Wait, \(78 - 3=75\), and \(45 - 3 = 42\). Wait, maybe I misread the available area. Wait the problem says 48 - foot by 81 - foot rectangular area. So width = 48 ft, length = 81 ft.

So pool width: \(48-2\times3 = 42\) ft (subtract 3 ft from left and 3 ft from right).
Pool length: \(81-2\times3=75\) ft (subtract 3 ft from top and 3 ft from bottom). Wait but in the diagrams, the y - coordinate of the top of the available area in the last diagram is 81, and the pool top is 78, so \(81 - 3=78\), and pool bottom is 3, so \(3-0 = 3\), so length of pool is \(78 - 3=75\), and width is \(45 - 3 = 42\) (since \(45-3 = 42\), and \(48 - 3=45\) on the right).

Step2: Calculate Perimeter of pool

Perimeter of a rectangle is \(P = 2\times(\text{length}+\text{width})\)
Substitute length = 75 ft and width = 42 ft:
\(P=2\times(75 + 42)=2\times117 = 234\) ft

Step3: Calculate Area of pool

Area of a rectangle is \(A=\text{length}\times\text{width}\)
Substitute length = 75 ft and width = 42 ft:
\(A = 75\times42=3150\) square feet

Answer:

Perimeter: \(\boldsymbol{234}\) feet, Area: \(\boldsymbol{3150}\) square feet