Question

A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope. Find the area of that part of the field in which the horse can graze. Also, find the increase in grazing area if length of rope is increased to 10 m. (Use π = 3.14)

Answer 1

Given:

Square field with side = 15 m.

Rope length initially r₁ = 5 m, then r₂ = 10 m.

Rope is tied at a corner, so the horse can graze a quarter 90° sector of a circle of radius r.

Step-wise calculation:

1. Formula for area of a quarter circle:

Area = `(1/4)·π·r^2`

Use π = 3.14

2. For r₁ = 5 m:

Area₁ = `(1/4)·3.14·(5)^2`

= `(1/4)·3.14·25`

= 3.14·6.25

= 19.625 m²

3. For r₂ = 10 m:

Area₂ = `(1/4)·3.14·(10)^2`

= `(1/4)·3.14·100`

= 3.14·25

= 78.50 m²

4. Increase in grazing area = Area₂ – Area₁

= 78.50 – 19.625

= 58.875 m²

Grazing area with 5 m rope = 19.625 m².

Grazing area with 10 m rope = 78.50 m².

Increase in grazing area = 58.875 m².