Surveying a Triangular Garden Plot
A landscape architect is surveying a triangular garden plot ABC for a client. She measures side AB as 24.0 m and side AC as 18.0 m, with the included angle ∠BAC equal to 52°. The client wants to build a fence along the boundary BC and to lay turf over the entire plot. The fencing contractor charges €45 per metre, and turf costs €8.50 per square metre. The architect also needs to know the perpendicular distance from vertex A to the side BC, because a drainage channel will run from A to the opposite side. A neighbouring property boundary runs parallel to BC at a distance of 9.0 m from BC on the same side as A; the architect must confirm that the plot does not cross that boundary. She uses the cosine rule, the area formula, and trigonometric ratios. All measurements are in metres and angles in degrees.
1. Calculate the length of BC and the area of the triangular plot.
Read the worked answer
Length BC by the cosine rule:
[
BC^2=AB^2+AC^2-2(AB)(AC)\cos(\angle BAC).
]
Substitute:
[
BC^2=24^2+18^2-2(24)(18)\cos52^\circ.
]
Now (24^2=576), (18^2=324), and (2(24)(18)=864). Also (\cos52^\circ\approx0.61566). Thus
[
BC^2=576+324-864(0.61566)
=900-531.93\approx368.07.
]
So
[
BC\approx\sqrt{368.07}\approx19.18\text{ m}.
]
Area using the included angle:
[
\text{Area}=\frac12(AB)(AC)\sin(\angle BAC)
=\frac12(24)(18)\sin52^\circ.
]
Since (\sin52^\circ\approx0.78801),
[
\text{Area}=216(0.78801)\approx170.21\text{ m}^2.
]
So (BC\approx19.18) m and the area is about (170.2) m².
2. Find the perpendicular distance from A to BC. Hence determine whether the plot crosses the neighbouring boundary that is 9.0 m from BC on the same side as A.
Read the worked answer
Let (h) be the perpendicular distance from A to BC. The area can also be written as
[
\text{Area}=\frac12(BC)(h).
]
Using the area from the previous task:
[
170.21=\frac12(19.18)h.
]
So
[
h=\frac{2(170.21)}{19.18}\approx\frac{340.42}{19.18}\approx17.75\text{ m}.
]
The perpendicular distance from A to BC is about 17.75 m.
The neighbouring boundary is parallel to BC and 9.0 m from BC on the same side as A. Since (17.75>9.0), the plot extends more than 9.0 m from BC towards A. Therefore the plot does cross the 9.0 m boundary line, so it does encroach on the neighbouring boundary. The architect must report this and adjust the design.
3. Calculate the cost of fencing BC and turfing the plot. If the client's budget is €2,000, is it sufficient?
Read the worked answer
Fencing cost:
[
19.18\text{ m}\times €45/\text{m}\approx €863.10.
]
Turfing cost:
[
170.21\text{ m}^2\times €8.50/\text{m}^2\approx €1{,}446.79.
]
Total cost:
[
863.10+1{,}446.79\approx €2{,}309.89.
]
Rounded sensibly, the total is about €2,310. The budget of €2,000 is therefore not sufficient; the client is short by about (2{,}310-2{,}000=€310). The client would need to increase the budget or reduce the scope, for example by using a cheaper turf or fencing option.