The School Garden Fence Problem
Students are fencing a rectangular garden against a wall. The wall forms one long side, so fencing is needed on only three sides. They have 24 metres of fencing. One student, Ari, says the largest area comes from a 6 m by 6 m square because 'a square always gives the most area.' Another student, Bea, draws a plan 8 m wide (parallel to the wall) and 8 m deep, using 8 + 8 + 8 = 24 m of fence. A third plan is 12 m wide and 6 m deep, using 12 + 6 + 6 = 24 m. The class must choose the plan that encloses the greatest area, and the garden must be at least 5 m deep for the raised beds.
1. Compute the area of each of the three plans and identify which uses the 24 m of fencing correctly.
Read the worked answer
Remember only three sides are fenced: two depths and one width.
- Ari's 6 by 6: fence used = 6 + 6 + 6 = 18 m, not 24. Area = 36 m². This plan does not use all the fencing, so it is not a fair comparison.
- Bea's 8 wide, 8 deep: fence = 8 + 8 + 8 = 24 m ✓. Area = 8 × 8 = 64 m².
- Third plan, 12 wide, 6 deep: fence = 12 + 6 + 6 = 24 m ✓. Area = 12 × 6 = 72 m².
The 12 by 6 plan encloses more area (72 m²) than the 8 by 8 plan (64 m²). Ari's square idea fails because with a wall on one side the shape is not symmetric; the best rectangle is not a square.
2. Let the depth be d metres. Write an expression for the area in terms of d, and use it to find the depth that gives the greatest area while keeping d at least 5 m.
Read the worked answer
The two depths use 2d metres of fence, so the width is 24 − 2d. Area = d(24 − 2d) = 24d − 2d².
Test whole-number depths (width must be positive, so d < 12):
- d = 5: width 14, area 70
- d = 6: width 12, area 72
- d = 7: width 10, area 70
- d = 8: width 8, area 64
The area peaks at d = 6, giving width 12 and area 72 m². Depths below 5 m are not allowed by the bed constraint, and depths above 6 shrink the area. So the 12 m by 6 m plan is best, confirming the earlier comparison and showing why the square guess was wrong.