Question

A farmer wants to purchase a triangular-shaped land with sides 120 feet and 60 feet and the angle included between these two sides is 60°. If the land costs Rs.500 per square feet, find the amount he needed to purchase the land. Also, find the perimeter of the land

Answer 1

Let ∆ABC be the shape of the land.
Given AB = 120 ft, AC = 60 ft
∠BAC = 60°
Using cosine formula in ∆ABC
BC² = AB² + AC² – 2AB . AC cos ∠BAC
BC² = 120² + 60² – 2 × 120 × 60 cos(60°)
= `144000 + 36000 - 14000 xx 1/2`
= 18000 – 7200
BC² = 10800 = 100 × 2 × 2 × 3 × 3 × 3
BC² = 10² × 2² × 3² × 3
BC = `sqrt(10^2 xx 2^2 xx 3^2 xx 3)`
BC = `10 xx 2 xx 3sqrt(3)`
BC = `60sqrt(3)` k.m.
Perimeter of the Land = AB + BC + AC
= `120 + 60sqrt(3) + 60`
= `180 + 60sqrt(3)`
= `60(3 + sqrt(3))` feet.
Area of ∆ABC = `1/2 xx "AB" xx "AC" xx sin∠"BAC"`
= `1/2 xx 60 xx 120  sin 60^circ`
= `30 xx 120 xx sqrt(3)/2`
= `30 xx 60 xx sqrt(3)`
= `1800 sqrt(3)` sq.feet
Cost of 1 sq.feet Rs.500
∴ Cost of `800  sqrt(3)` sq.feet = `800  sqrt(3) xx 500`
= `900000 sqrt(3)`
Total amount needed = `900000 sqrt(3)`
Perimeter of the land = `60(3 + sqrt(3))` feet.