A cow is tied with a rope 14 m long at the corner of a rectangular field of dimensions 20 m × 16 m. Find the area it can graze.
\(49\pi\;\text{m}^2\) (≈ 153.94 m²)
Step 1: Understand the situation.
A cow is tied at the corner of a rectangular field with a rope of length 14 m. This means the cow can move around only in a circular path of radius 14 m, but since it is tied at a corner, it cannot move in all directions — it can move only in one-quarter (1/4) of the circle.
Step 2: Check the field size.
The field has length 20 m and breadth 16 m. Both of these are larger than 14 m (the rope length). So the cow’s full circular movement of radius 14 m fits inside the field boundaries at the corner.
Step 3: Area grazed is a quarter of a circle.
When tied at a corner, the grazed area is a quarter-circle with radius \(r = 14\,\text{m}\).
Step 4: Formula for area of a circle.
The full area of a circle is given by: \(A = \pi r^2\)
Step 5: Quarter of the circle.
Since only 1/4th of the circle is available: \(A = \dfrac{1}{4} \pi r^2\)
Step 6: Substitute values.
\(A = \dfrac{1}{4} \pi (14^2)\)
\(A = \dfrac{1}{4} \pi (196)\)
\(A = 49\pi\;\text{m}^2\)
Step 7: Approximate value.
Using \(\pi ≈ 3.1416\), \(A ≈ 49 × 3.1416 = 153.94\,\text{m}^2\).
Final Answer: The cow can graze an area of \(49\pi\;\text{m}^2\) (≈ 153.94 m²).