Question

A horse is tied to a peg at one corner of square shaped grass field of side 15 m by means of 5 m long rope, find

1. The area of that part of the field in which the horse can graze. 
2. The increase in the grazing area if the rope were 10 m long instead of 5 m. (π = 3.14)

Answer 1

Given:

Square field side = 15 m.

Rope length (a) = 5 m, (b) = 10 m.

Horse tied at one corner so accessible region inside the field is a quarter-circle (sector of 90°).

Use π = 3.14.

Step-wise calculation:

1. Formula: Area of a sector = `θ/(360^circ ) xx π xx r^2`. 

For a quarter-circle θ = 90°, so area = `1/4 πr^2`.

2. For rope = 5 m:

r = 5 → Area = `1/4 xx π xx 5^2`

= `25/4 xx π`

= 6.25π

Numeric: 6.25 × 3.14 = 19.625 m²

3. For rope = 10 m:

r = 10 → Area = `1/4 xx π xx 10^2`

= `100/4 xx π`

= 25π

Numeric: 25 × 3.14 = 78.50 m²

4. Increase when rope becomes 10 m instead of 5 m:

Increase = 78.50 – 19.625

= 58.875 m²

a. Grazing area with 5 m rope = 6.25π = 19.625 m².

b. Increase if rope were 10 m instead = 18.75π = 58.875 m².