The Garden Gate and the Wall Limit
At a community-center meeting, Nadia is designing a rectangular garden. She has 40 feet of fencing, and a 4-foot gate will be set into one of the fenced sides; the gate replaces fencing, so the total boundary is 44 feet. One side of the garden runs along a garage wall, and that side cannot be longer than 10 feet because of a downspout. Nadia wants whole-foot dimensions and the greatest possible area. She sketches an 11-by-11 square first, then tries 12 by 10 and 14 by 8. A volunteer says the square always gives the most area for a fixed perimeter. Nadia is not sure the square is allowed. The group needs a recommendation, a check of at least three possible dimensions, and a sentence explaining why the best choice is not the square.
1. Write an equation relating the fenced dimensions, then decide whether 11 by 11 is allowed. Show the check.
Read the worked answer
Let the two dimensions be L and W. The gate replaces 4 feet of fencing, so the fencing plus the gate equals the perimeter:
2(L + W) = 40 + 4 = 44, so L + W = 22.
The side along the garage cannot exceed 10 feet. If that side is W, then W ≤ 10.
An 11-by-11 square has W = 11, which is greater than 10. It is not allowed, even though it satisfies L + W = 22. The volunteer's 'square always wins' rule applies only when no side limit prevents the square. Here the wall limit rules out the square.
2. Test at least three whole-foot pairs that satisfy the wall limit, then recommend the dimensions. Explain why the square would win if the wall limit were removed.
Read the worked answer
Using L + W = 22 and W ≤ 10:
- 12 by 10: area = 120 square feet (W = 10, allowed)
- 13 by 9: area = 117 square feet
- 14 by 8: area = 112 square feet
A fourth check, 10 by 12, is the same as 12 by 10 and also gives 120 square feet.
Recommendation: 12 feet by 10 feet, with the 10-foot side along the garage wall. This gives the greatest allowed area, 120 square feet.
If the wall limit were removed, the best whole-foot dimensions would be 11 by 11, giving 121 square feet. The square is best for a fixed perimeter, but it violates the wall limit by 1 foot. The best allowed rectangle is therefore the closest allowed shape to the square.
3. If the wall limit were raised to 11 feet, what would the best dimensions be, and by how much would the area increase?
Read the worked answer
With a limit of 11 feet, the 11-by-11 square is now allowed. It still satisfies L + W = 22 and gives:
11 × 11 = 121 square feet
Compared with the previous best of 120 square feet, the increase is:
121 − 120 = 1 square foot
So the best dimensions become 11 by 11, and the area rises by only 1 square foot. This small change makes sense because the square was already just 1 foot beyond the old limit.