Fencing the Garden
The fourth-grade garden club has 24 feet of fencing to enclose a rectangular garden. They must use all 24 feet and want the largest possible area. The garden must have whole-number side lengths in feet. One student, Priya, says a 6 ft by 6 ft square uses 24 ft and has area 36 sq ft, so nothing can be larger. Another student, Marcus, says a long skinny rectangle like 1 ft by 11 ft also uses 24 ft, so the area should be 11 sq ft, which is smaller. The club has grid paper and wants a table of all possible rectangles. They also need to explain why the square gives the greatest area among the whole-number choices, not just assert it. The teacher asks you to help the club present a convincing argument.
1. List all whole-number dimensions of rectangles with perimeter 24 ft, and compute each area.
Read the worked answer
Perimeter 24 means 2(L + W) = 24, so L + W = 12. The whole-number pairs with L ≥ W are:
| Dimensions | Area |
|---|---|
| 1 ft × 11 ft | 11 sq ft |
| 2 ft × 10 ft | 20 sq ft |
| 3 ft × 9 ft | 27 sq ft |
| 4 ft × 8 ft | 32 sq ft |
| 5 ft × 7 ft | 35 sq ft |
| 6 ft × 6 ft | 36 sq ft |
The 6 ft by 6 ft square has the greatest area in the table.
2. Explain why the 6 ft by 6 ft square has the greatest area, using the table and the idea that when the sides are closer together the area is larger.
Read the worked answer
With L + W fixed at 12, the product L × W is largest when L and W are as close as possible. Moving from 6 × 6 to 5 × 7 changes the area from 36 to 35; moving farther to 4 × 8 gives 32, then 27, 20, and 11. Each step makes the rectangle longer and thinner, so area drops. This is not a proof for all real-number side lengths, but it is a complete check for whole-number side lengths, which is the stated constraint.
3. Suppose the club wants an area of at least 30 sq ft. Which dimensions from the table work?
Read the worked answer
From the table, the areas of at least 30 sq ft are 4 × 8 = 32, 5 × 7 = 35, and 6 × 6 = 36. The 3 × 9 = 27 and smaller rectangles do not meet the goal. So the club could choose any of those three whole-number rectangles and still have at least 30 sq ft; the square still gives the largest area.