Fencing a Square Garden with an Irrational Side Length
The garden club at Riverside Middle School has permission to build a square plot against the east wall of the courtyard. The plot must have an area of exactly 50 square meters, and the club has already ordered soil for that area. Fencing is sold only in rigid 1-meter sections; sections cannot be cut, bent, or overlapped, and each of the four sides must be fully covered. A volunteer measured with a calculator, wrote down 7.07 m as the side length, and told the club to 'just round to 7.1 and buy 7 panels per side, since 7 is close enough.' The treasurer is worried because the soil order and the fence order must both be correct before Saturday, and returning either is expensive. A second design is also on the table: a rectangular plot 7 m by 8 m, which a parent says 'uses less fence but holds more soil.' The club needs a decision it can defend with arithmetic rather than estimates.
1. Show, by squaring, whether a side length of 7.1 m gives the required area, and determine whether 7.07 m is an overestimate or underestimate of the exact side length. State the error in area for each.
Read the worked answer
Square each candidate and compare with 50.
- 7.1² = 50.41, which is 0.41 m² too large, so 7.1 m is too long.
- 7.07² = 49.9849, which is 0.0151 m² too small, so 7.07 m is too short.
Because 7.07² < 50 < 7.1², the exact side length lies between 7.07 and 7.1: 7.07 < √50 < 7.1. So 7.07 underestimates √50. The exact value is √50 = 5√2 ≈ 7.0710678 m, which is irrational, so no finite decimal is exact. Rounding 7.07 to 7.1 changes the area by about 0.43 m² relative to 49.9849, which is enough to matter when soil and fencing are ordered.
2. Fence sections are 1 m long and cannot be cut. How many sections are needed per side, and what is the total number of sections? How much fence is unused or missing? Justify the rounding decision.
Read the worked answer
The side length is √50 ≈ 7.071 m, which is greater than 7 m. Using 7 sections per side leaves a gap of about 0.071 m on each side, so the plot is not fully enclosed; the sections cannot be stretched or overlapped, so rounding down is not allowed. Round up to 8 sections per side.
Total: 8 × 4 = 32 sections, or 32 m of fencing.
The perimeter actually needed is 4√50 ≈ 28.284 m. The excess is 32 − 28.284 = 3.716 m, which is unavoidable with rigid 1-m sections. The volunteer's plan of 7 per side would give only 28 m of fence, leaving four gaps of about 7 cm each.
3. Evaluate the parent's claim that the 7 m by 8 m rectangle holds more soil but uses less fence. Support the answer with area and perimeter calculations, and note any condition under which the rectangle is not acceptable.
Read the worked answer
Area of the rectangle: 7 × 8 = 56 m², compared with 50 m² for the square, so it holds 6 m² more soil. The claim about soil is true.
Fence: the perimeter is 2(7 + 8) = 30 m, so 30 sections are needed, compared with 32 for the square. Because 7 m and 8 m are whole numbers of meters, each side is covered exactly with no waste. The claim about using less fence is also true.
So both parts of the claim are correct. However, the rectangle is not a square, so it is unacceptable if the design requires a square plot. The rectangle is also more efficient: 30 ÷ 56 ≈ 0.54 sections per square meter, versus 32 ÷ 50 = 0.64 for the square.