NCERT Class X Chapter 11: Area Related To Circles Exercise 11.1 Question 8

NCERT Class X Chapter 11: Area Related To Circles

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.
(i) Find the area of that part of the field in which the horse can graze.
(ii) Find the increase in the grazing area if the rope were 10 m long instead of 5 m.
(Use π = 3.14)

Given:

  • Side of square field = 15 m
  • Length of rope = 5 m (initially), then 10 m
  • π = 3.14

To Find:

  • (i) Area of field in which horse can graze with 5 m rope
  • (ii) Increase in grazing area if rope length is increased to 10 m

Formula:

  • Area of a circle:   \( A = \pi r^2 \)
  • Area of a quarter circle:   \( A = \dfrac{\pi r^2}{4} \)

Solution:

Step 1: Area grazed when rope is 5 m

The horse grazes a quarter circle of radius 5 m.

$$ \text{Area}_1 = \frac{\pi r^2}{4} = \frac{3.14 \times (5)^2}{4} $$

Step 2: Calculate the value for 5 m rope

$$ \text{Area}_1 = \frac{3.14 \times 25}{4} = \frac{78.5}{4} = 19.625 \ \text{m}^2 $$

Step 3: Area grazed when rope is 10 m

The horse grazes a quarter circle of radius 10 m.

$$ \text{Area}_2 = \frac{\pi r^2}{4} = \frac{3.14 \times (10)^2}{4} $$

Step 4: Calculate the value for 10 m rope

$$ \text{Area}_2 = \frac{3.14 \times 100}{4} = \frac{314}{4} = 78.5 \ \text{m}^2 $$

Step 5: Increase in grazing area

$$ \text{Increase} = \text{Area}_2 - \text{Area}_1 = 78.5 - 19.625 = 58.875 \ \text{m}^2 $$

Result:

  • (i) Area the horse can graze with 5 m rope = 19.625 m2
  • (ii) Increase in grazing area if rope is 10 m = 58.875 m2
© Kaliyuga Ekalavya. All rights reserved.

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