Community garden: composite area, topsoil volume and fencing quantities
The Kaiti community garden has been given a corner of reserve land. The plot is a rectangle 6.4 m by 3.8 m with a semicircular end on one of the 3.8 m sides. The whole plot is to be edged, fenced and filled with topsoil to a settled depth of 250 mm.
Topsoil is sold in 40 L bags at $8.90 each, with $75 delivery per order. The supplier gives 12% off the goods (but not the delivery) when the goods total is over $1,500. The committee budgets 10% more topsoil than the calculated volume, because soil settles and the ground is slightly uneven.
Fencing is supplied in 1.8 m wide panels at $34.50 each, with a 1.0 m wide gate. Posts and concrete have already been donated. One committee member has costed the job at 180 bags, saying "close enough, it's only a few centimetres". Another wants 250 bags "to be safe".
1. Calculate the area of the plot and the number of 40 L bags of topsoil needed, including the 10% allowance. Round sensibly and explain the rounding.
Read the worked answer
Area
- Rectangle: 6.4 × 3.8 = 24.32 m²
- Semicircle: diameter 3.8 m, so radius 1.9 m. Area = ½ × π × 1.9² = ½ × π × 3.61 = 5.67 m²
- Total area = 24.32 + 5.67 = 29.99 m² (call it 30.0 m²)
Volume at 250 mm = 0.25 m
- 30.0 × 0.25 = 7.50 m³ = 7,500 L
With the 10% settling allowance
- 7,500 × 1.10 = 8,250 L
- Bags = 8,250 ÷ 40 = 206.25 → 207 bags
Round up, never down: 206 bags leaves 8,240 L, which is below the allowance, and you cannot buy a quarter bag. Keeping the area as 29.99 m² rather than rounding to 30 gives 206.2 bags - the same order of 207, which shows the answer is robust to that rounding. The 10% allowance already covers the uneven ground, so it should not be applied twice.
2. Calculate the number of fence panels required and the total cost of the topsoil and fencing, applying the discount correctly.
Read the worked answer
Fencing length
- Three straight sides: 6.4 + 6.4 + 3.8 = 16.6 m
- Semicircular arc: ½ × π × 3.8 = 5.97 m
- Total perimeter = 22.57 m
- Less the 1.0 m gate = 21.57 m to fence
- Panels = 21.57 ÷ 1.8 = 11.98 → 12 panels (the twelfth is cut down; you still pay for it)
Topsoil cost
- Goods: 207 × $8.90 = $1,842.30. This exceeds $1,500, so the 12% discount applies.
- Discounted goods = $1,842.30 × 0.88 = $1,621.22
- Add delivery (not discounted): $1,621.22 + $75.00 = $1,696.22
Fencing cost
- 12 × $34.50 = $414.00
Total = $1,696.22 + $414.00 = $2,110.22
Discount trap: applying 12% to the delivery as well gives $1,614.32 - $6.90 too low. Apply the discount only to the goods subtotal, as the supplier's terms state.
3. Evaluate the two suggestions - 180 bags and 250 bags - against the 250 mm settled depth requirement. Give a recommendation with a reason.
Read the worked answer
180 bags
- 180 × 40 = 7,200 L = 7.2 m³
- Depth = 7.2 ÷ 30.0 = 0.24 m = 240 mm
- That is 10 mm short before any settling. After settling (the reason for the 10% allowance) the working depth is about 240 × 0.9 = 216 mm, roughly 34 mm under specification.
250 bags
- 250 × 40 = 10,000 L = 10 m³
- Depth = 10 ÷ 30.0 = 333 mm
- That is 83 mm over, about 1.8 m³ of extra soil, roughly $400 wasted, and the deeper fill will settle unevenly and may bury the edging.
Recommendation: 207 bags. It meets the 250 mm settled depth with a deliberate 10% margin rather than a guess. The "close enough" argument fails because the shortfall compounds with settling, and the garden beds would end up visibly shallow; the "to be safe" argument fails on cost. Neither suggestion is based on the calculated volume, so both should be rejected.