Pond and path: forming and solving a quadratic
A landscape gardener is designing a rectangular pond and a surrounding paved path for a walled courtyard. The pond is to be 8 m long and 5 m wide, and the path will be a uniform width of x metres on all four sides. The client has asked for the pond and path together to cover exactly 70 m², and has a budget of £1,500 for paving at £46 per square metre. During a site visit the gardener discovers that the path must be at least 1.2 m wide so that a wheelchair can turn safely at the corner, and that the courtyard walls cannot be moved. The client is keen to keep the 8 m by 5 m pond if at all possible. The gardener must decide whether the original brief can be met and, if it cannot, prepare two realistic alternatives with figures to discuss at the next meeting.
1. Form and solve an equation to find the path width x, giving the answer to a sensible degree of accuracy and explaining why one solution is rejected.
Read the worked answer
The overall rectangle is (8 + 2x) m by (5 + 2x) m, because the path adds x at each end of each side.
Set the total area equal to 70:
(8 + 2x)(5 + 2x) = 70 40 + 16x + 10x + 4x² = 70 4x² + 26x − 30 = 0 2x² + 13x − 15 = 0
Use the quadratic formula with a = 2, b = 13, c = −15:
x = [−13 ± √(169 + 120)] ÷ 4 = [−13 ± √289] ÷ 4 = (−13 ± 17) ÷ 4
x = 1 or x = −7.5
Reject x = −7.5 because a length cannot be negative. So the path is 1 m wide.
Check: 10 × 7 = 70 m², correct.
The reasoning mark is for justifying the rejection of the negative root physically, not just algebraically.
2. Using x = 1, calculate the area of the path and its cost, and say whether the client's £1,500 budget is enough.
Read the worked answer
Area of the pond = 8 × 5 = 40 m².
Total area = 70 m², so the path area is 70 − 40 = 30 m².
Alternatively, the overall rectangle is 10 m by 7 m, area 70 m², and subtracting the pond gives the same 30 m².
Cost = 30 × £46 = £1,380.
This is within the £1,500 budget, leaving £120 spare.
A common slip is to multiply the whole 70 m² by £46, which gives £3,220 and wrongly suggests the budget fails. Only the paved strip is being paid for.
3. Explain why the design cannot now go ahead, using calculations. Give two realistic alternatives with figures.
Read the worked answer
The path must be at least 1.2 m wide, but the area condition forces x = 1, so the two requirements conflict.
Try x = 1.2 with the pond still 8 m by 5 m:
- Overall dimensions: 8 + 2.4 = 10.4 m and 5 + 2.4 = 7.4 m
- Total area: 10.4 × 7.4 = 76.96 m² (not 70 m²)
- Path area: 76.96 − 40 = 36.96 m²
- Cost: 36.96 × £46 = £1,700.16, which is £200.16 over budget
So with the walls fixed and the pond unchanged, the brief is impossible.
Two defensible alternatives:
- Keep the 8 m by 5 m pond and the 1.2 m path, raise the paving budget to about £1,701 and accept a total area of 76.96 m².
- Keep the 70 m² total area and a 1.2 m path, but shrink the pond. For example, an overall rectangle of 10 m by 7 m with a 1.2 m path gives a pond of 7.6 m by 4.6 m, which satisfies both conditions.
A third option, if the client insists on the pond and the budget, is a 1 m path with a widened 1.5 m turning pad at one corner, but that needs a separate design check.