Question

A horse, a cow and a goat are tied, each by ropes of length 14 m, at the corners A, B and C respectively, of a grassy triangular field ABC with sides of lengths 35 m, 40 m and 50 m. Find the area of grass field that can be grazed by them.

Answer 1

Each animal is tied at a vertex of triangle ABC.

Horse at A.

Cow at B.

Goat at C.

Each rope has a length of 14 m, so each animal can graze a sector of a circle of radius 14 m.

But the angle of each grazing sector is equal to the interior angle of the triangle at that vertex.

Let the angles of triangle ABC be:

∠A = θ₁, ∠B = θ₂, ∠C = θ₃

We know that:

θ₁ + θ₂ + θ₃ = 180°

Area grazed at each corner:

Area of a sector of radius 14 m and angle θ°:

Sector Area = `θ/(360°)pir^2`

Total grazed area = `θ_1/360^circ xx π(14)^2 + θ_2/360^circ xx π(14)^2 + θ_3/360^circ xx π(14)^2`

Factor out common terms:

= `(π(14)^2)/360^circ (θ_1 + θ_2 + θ_3)`

= `(π(14)^2)/360^circ xx 180^circ`

= `1/2pi(14)^2`

Grazing area = `1/2pi(14)^2`

= `1/2 xx pi xx 196`

= `1/2 xx 22/7 xx 196   ...("Using"  pi = 22/7)`

= `1/2 xx 22 xx 28`

= 11 × 28

= 308 m²