Garden Fence and Soil Cost Design
The garden club has 48 feet of fencing to enclose a rectangular vegetable plot against the school wall. The wall forms one full side, so fencing is needed for the other three sides. The club wants the plot to be at least 200 square feet and the side parallel to the wall to be no more than 20 feet. A 2-foot-wide walkway will run along the wall inside the plot, reducing planted area. Soil costs $6.50 per cubic foot and must fill the planted area to a depth of 0.5 foot. The club must decide the dimensions before ordering soil. A member suggests making the side parallel to the wall 16 feet. The adviser asks whether that leaves enough planted area and what dimensions would maximize area under the constraints.
1. Let x be the side parallel to the wall and y the depth. Write the fencing equation and find the dimensions that maximize total area under the x ≤ 20 constraint.
Read the worked answer
Fencing covers three sides: x + 2y = 48, so y = (48 − x)/2.
Area A = xy = x(48 − x)/2 = 24x − x^2/2.
This is a downward parabola with vertex at x = 24, where y = 12 and A = 288. But the constraint says x ≤ 20, and the area increases as x goes from 0 to 24. So the constrained maximum is at x = 20. Then y = (48 − 20)/2 = 14.
Maximum allowed total area = 20 × 14 = 280 square feet.
2. Test the member’s proposal x = 16. Does the planted area after the 2-foot walkway meet the 200-square-foot requirement?
Read the worked answer
If x = 16, y = (48 − 16)/2 = 16. Total area = 16 × 16 = 256 square feet.
The walkway runs along the wall inside the plot, so it removes a 2-foot strip from the depth: planted depth = 16 − 2 = 14. Planted area = 16 × 14 = 224 square feet.
This meets the 200-square-foot minimum with 24 square feet to spare. The proposal is feasible, but it does not maximize total area; x = 20 gives total 280 and planted 20(14 − 2) = 240 square feet.
3. Find the soil cost for the x = 16 design and compare it with the cost for the x = 20 design.
Read the worked answer
For x = 16, planted volume = 224 × 0.5 = 112 cubic feet. Cost = 112 × $6.50 = $728.
For x = 20, planted area = 20(14 − 2) = 240 square feet; volume = 240 × 0.5 = 120 cubic feet; cost = 120 × $6.50 = $780.
The x = 20 design costs $52 more but provides 16 more square feet of planted area and the maximum total area allowed. Compare cost per planted square foot: $728/224 = $3.25 per square foot; $780/240 = $3.25 per square foot. Soil cost per planted square foot is the same, so the larger design is not less efficient; it simply uses more soil.