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

Given: Triangle \(ABC\) with side lengths \(35\,\text{m}, 40\,\text{m}, 50\,\text{m}\). A horse, cow and goat are tied at \(A,B,C\) respectively with equal rope length \(r=14\,\text{m}\). Animals graze only inside the field.

Key idea

At each vertex, the animal grazes a circular sector of radius \(r\) subtended by the triangle’s interior angle at that vertex. Hence, total grazed area \[ \mathcal{A} \;=\; \frac{1}{2}r^2(\angle A+\angle B+\angle C). \] But in any triangle, \(\angle A+\angle B+\angle C=\pi\) radians \(\Rightarrow\) \[ \mathcal{A} \;=\; \frac{1}{2}\,r^2\,\pi. \] Here \(r=14\,\text{m}\), and each adjacent side at a vertex is longer than \(14\) m, so no sector is truncated by hitting the opposite side.

Compute

\[ \mathcal{A}=\frac{1}{2}\times 14^2 \times \pi \;=\; \frac{1}{2}\times 196 \times \pi \;=\; 98\pi\ \text{m}^2 \;\approx\; 307.88\ \text{m}^2. \]

Total grazed area: \(\boxed{98\pi\ \text{m}^2 \;(\approx 307.9\ \text{m}^2)}\).

Answer 2

Step 1: Understand the problem:
We are given a triangular field \( ABC \) with sides of lengths 35 m, 40 m, and 50 m. A horse, a cow, and a goat are tied to the corners \( A \), \( B \), and \( C \) of the triangle, respectively, with ropes of length 14 m each. We are asked to find the total area of the field that can be grazed by them.

Step 2: Identify the problem as an area of circle segments:
Each animal can graze an area in the form of a circular segment, with a radius of 14 m, from the point where it is tied. These circular segments are part of the full circle, and we need to calculate their total area.

Step 3: Calculate the area of the triangle ABC:
To find the total area of the triangle, we can use Heron's formula. First, we find the semi-perimeter \( s \) of the triangle:
\[ s = \frac{35 + 40 + 50}{2} = 62.5 \, \text{m} \] Now, we can use Heron's formula to find the area \( A \) of the triangle:
\[ A = \sqrt{s(s - a)(s - b)(s - c)} \] where \( a = 35 \), \( b = 40 \), and \( c = 50 \). Substituting the values:
\[ A = \sqrt{62.5(62.5 - 35)(62.5 - 40)(62.5 - 50)} = \sqrt{62.5 \times 27.5 \times 22.5 \times 12.5} \] \[ A = \sqrt{62.5 \times 27.5 \times 22.5 \times 12.5} = 600 \, \text{m}^2 \]

Step 4: Calculate the area of the circular segments:
Each animal can graze a circular area with a radius of 14 m. The area of the full circle is:
\[ \text{Area of full circle} = \pi r^2 = \pi (14)^2 = 196\pi \, \text{m}^2 \] However, since the animals are tied to the corners of the triangle, they can only graze part of the circle. The grazed area is a circular segment. To find the area of the circular segments, we need to calculate the central angle formed by the radius and the sides of the triangle.

Step 5: Calculate the central angles:
The central angle \( \theta \) at each corner of the triangle can be found using the formula for the area of a triangle in terms of the sides and the angle between them: \[ A = \frac{1}{2} r^2 \theta \] However, a more straightforward way is to calculate the total area of the three circular segments formed by the grazed area and sum them up. The angles depend on the specific geometry of the triangle, which requires using trigonometry.

Step 6: Calculate the total grazed area:
In a more simplified approach, we can estimate the total grazed area based on the geometry. The total grazed area is the area of three segments, and we add them together to get the total. If a more detailed geometric approach is used, the total grazed area could be calculated.

Conclusion:
The total grazed area would be the sum of the three circular segments formed by each animal. To calculate the exact total area, further geometric analysis is needed to account for the exact angles formed at the corners, but the general approach follows the above steps.