Radhey can walk along the boundary of a rectangular field and also along the diagonals of the field. His speed is 53 km/h. The length of the field is 45 km. Radhey started from one corner and reached to the diagonally opposite corner in 1 hour. What is the area of the field?

Options

A. `860 km^2`
B. `1260 km^2`
C. `1060 km^2`
D. can't be determined

Answer

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

### Step 1: Understand the Problem
Radhey walks diagonally across a rectangular field, starting from one corner to the diagonally opposite corner. We know his speed and the length of the field.

### Step 2: Identify Given Information
- Speed of Radhey = 53 km/h
- Length of the field (L) = 45 km
- Time taken to reach the opposite corner = 1 hour

### Step 3: Calculate the Distance Covered
Since Radhey takes 1 hour to reach the opposite corner, we can calculate the distance he covers using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
\[ \text{Distance} = 53 \text{ km/h} \times 1 \text{ h} = 53 \text{ km} \]

### Step 4: Use the Pythagorean Theorem
In a rectangle, the diagonal (D) can be found using the Pythagorean theorem:
\[ D = \sqrt{L^2 + W^2} \]
Where:
- L = Length of the rectangle = 45 km
- W = Width of the rectangle (unknown)

We know that the distance Radhey covered is equal to the diagonal:
\[ 53 = \sqrt{45^2 + W^2} \]

### Step 5: Square Both Sides
To eliminate the square root, we square both sides:
\[ 53^2 = 45^2 + W^2 \]
\[ 2809 = 2025 + W^2 \]

### Step 6: Solve for Width (W)
Now, we isolate W^2:
\[ W^2 = 2809 - 2025 \]
\[ W^2 = 784 \]
Taking the square root of both sides gives:
\[ W = \sqrt{784} = 28 \text{ km} \]

### Step 7: Calculate the Area of the Rectangle
The area (A) of a rectangle is given by:
\[ A = L \times W \]
Substituting the values we found:
\[ A = 45 \text{ km} \times 28 \text{ km} \]
\[ A = 1260 \text{ km}^2 \]

### Final Answer
The area of the field is **1260 km²**.

---