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 total area of grass field that can be grazed by them

Answer 1

Step 1: Understand the problem:
We are given that each animal can graze in a circular area with a radius of 14 m. The total area grazed by the three animals is the sum of the areas of the three circular regions they can graze in.

Step 2: Formula for the area of a circle:
The area $A$ of a circle is given by the formula: \[ A = \pi r^2 \] where $r$ is the radius of the circle and $\pi$ is a constant (approximately 3.1416).

Step 3: Area grazed by one animal:
Each animal can graze in a circle with a radius of 14 m. So, the area grazed by one animal is: \[ \text{Area grazed by one animal} = \pi (14)^2 = 196\pi \, \text{sq.m} \] Thus, the area grazed by each animal is $196\pi$ square meters.

Step 4: Total area grazed by all three animals:
Since there are three animals, the total area grazed by them is the sum of the areas grazed by each animal: \[ \text{Total area grazed} = 3 \times 196\pi = 588\pi \, \text{sq.m} \] Thus, the total area grazed by all three animals is $588\pi$ square meters.

Conclusion:
The total area grazed by the three animals is $588\pi$ square meters.