Area Grazed with 5m Rope: The horse can graze in a quarter circle area when tied with a 5m rope. The area is (1/4)×π×r², where r = 5 m. So, the area is (1/4)×3.14×5² = 19.625 m².
Area Grazed with 10m Rope: With a 10m rope, the horse can graze a larger quarter circle. The area is (1/4)×π×10² = 78.5 m².
Increase in Grazing Area: The increase is 78.5 − 19.625 = 58.875 m².
We can say that the grazing area increases significantly with the length of the rope.

Let’s discuss in detail

## Introduction to the Problem of Grazing Area

The problem at hand involves calculating the grazing area accessible to a horse tied to a peg at the corner of a square-shaped field. This scenario is a classic example of applying geometric concepts to real-world situations. The field has a side length of 15 meters, and the horse is initially tied with a 5-meter-long rope. The task is to determine the area available for grazing with this rope length and then assess how the grazing area changes if the rope’s length is increased to 10 meters. This problem not only tests our understanding of circle geometry but also illustrates how mathematical concepts can be applied to everyday situations.

### Calculating the Grazing Area with a 5-Meter Rope

When the horse is tied with a 5-meter rope, the area available for grazing is a quarter of a circle, with the radius equal to the length of the rope. The formula for the area of a circle is πr², but since the horse can only access a quarter of this circle, the formula becomes (1/4)×π×r². Substituting r = 5 meters and π = 3.14, the grazing area is calculated as (1/4)×3.14×5², which equals 19.625 square meters. This represents the area in which the horse can graze when restricted by a 5-meter rope.

#### Understanding the Increase in Grazing Area with a Longer Rope

The problem also asks us to consider the scenario where the length of the rope is doubled to 10 meters. Intuitively, one might expect the grazing area to also double, but the relationship between the radius of a circle and its area is quadratic, not linear. This means that increasing the rope length will have a more significant impact on the grazing area than a simple doubling. The new grazing area must be recalculated using the same quarter-circle formula but with the updated radius of 10 meters.

Calculating the Grazing Area with a 10-Meter Rope
With the rope length increased to 10 meters, the grazing area becomes a larger quarter circle. Using the formula (1/4)×π×r² with r = 10 meters, the new grazing area is calculated. Substituting these values gives us (1/4)×3.14×10², which equals 78.5 square meters. This significantly larger area demonstrates how a change in the radius of a circle dramatically affects its area, a fundamental concept in circle geometry.

##### Analyzing the Increase in Grazing Area

To find the increase in the grazing area due to the extension of the rope, we subtract the initial grazing area (with the 5-meter rope) from the new grazing area (with the 10-meter rope). The increase is therefore 78.5 − 19.625 = 58.875 square meters. This substantial increase highlights the quadratic relationship between the radius of a circle and its area. It shows that even a small increase in the radius (or in this case, the length of the rope) can lead to a significant increase in area.

###### Geometric Principles in Everyday Life

In conclusion, this problem illustrates the practical application of geometric principles in everyday life. By understanding the relationship between the radius of a circle and its area, we can solve real-world problems such as determining the grazing area for a horse. This example not only reinforces the importance of geometry in practical scenarios but also demonstrates how a seemingly simple change (like increasing the length of a rope) can have a significant impact on the outcome. It’s a testament to the power and relevance of mathematics in understanding and solving everyday challenges.

Discuss this question in detail or visit to Class 10 Maths Chapter 11 for all questions.
Questions of 10th Maths Exercise 11.1 in Detail

 Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°. Find the area of a quadrant of a circle whose circumference is 22 cm. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes. A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding (i) minor segment (ii) major sector. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find: (i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord. A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle. A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle. 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 (i) the area of that part of the field in which the horse can graze. (ii) the increase in the grazing area if the rope were 10 m long instead of 5 m. A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors. Find (i) the total length of the silver wire required. (ii) the area of each sector of the brooch. An umbrella has 8 ribs which are equally spaced. Assuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella. A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115°. Find the total area cleaned at each sweep of the blades. To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle 80° to a distance of 16.5 km. Find the area of the sea over which the ships are warned. A round table cover has six equal designs. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of ₹0.35 per cm².

Last Edited: June 11, 2024