A horse is tethered to one corner of a square plot of side 42 m by a 30 m long rope, then
(i) Find the area it can graze
(ii) Find the area that will be left ungrazed.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the Problem
A horse is tethered to one corner of a square plot with a side length of 42 m using a rope that is 30 m long. We need to find:
(i) The area the horse can graze.
(ii) The area that will be left ungrazed.

### Step 2: Calculate the Area the Horse Can Graze
The horse can graze in a quarter-circle area because it is tethered at one corner of the square. The radius of this quarter-circle is the length of the rope, which is 30 m.

**Area of a quarter-circle** is given by the formula:
\[ \text{Area} = \frac{1}{4} \pi r^2 \]
Where \( r = 30 \) m.

Substituting the value of \( r \):
\[ \text{Area} = \frac{1}{4} \times \frac{22}{7} \times (30)^2 \]
Calculating \( (30)^2 = 900 \):
\[ \text{Area} = \frac{1}{4} \times \frac{22}{7} \times 900 \]
\[ = \frac{22 \times 900}{4 \times 7} \]
\[ = \frac{19800}{28} \]
\[ = 707.14 \text{ m}^2 \]

### Step 3: Calculate the Total Area of the Square Plot
The area of the square plot is calculated using the formula:
\[ \text{Area of square} = \text{side}^2 \]
Where the side length is 42 m:
\[ \text{Area of square} = 42^2 = 1764 \text{ m}^2 \]

### Step 4: Calculate the Ungrazed Area
To find the area that will be left ungrazed, we subtract the grazed area from the total area of the square:
\[ \text{Ungrazed Area} = \text{Total Area} - \text{Grazed Area} \]
Substituting the values we calculated:
\[ \text{Ungrazed Area} = 1764 - 707.14 \]
\[ = 1056.86 \text{ m}^2 \]

### Final Answers
(i) The area the horse can graze is **707.14 m²**.
(ii) The area that will be left ungrazed is **1056.86 m²**.

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