Two goats tethered to diagonally opposite vertices of a field formed by joining the mid-points of the adjacent sides of another square field of side `20sqrt(2)` . What is the total grazing area of the two goats?

Options

A. `100 pi m^(2)`
B. `50 ( sqrt(2) -1 ) pi m^(2)`
C. `100 pi (3 -2sqrt(2)) m^(2)`
D. `200 pi ( 2-sqrt(2))m^(2)`

Answer

To solve the problem of finding the total grazing area of the two goats tethered to diagonally opposite vertices of a square field formed by joining the midpoints of the adjacent sides of another square field, we can follow these steps:

### Step 1: Understand the Geometry of the Field
The original square field has a side length of \(20\sqrt{2}\). When we join the midpoints of the adjacent sides, we form a smaller square. The vertices of this smaller square will be the midpoints of the sides of the larger square.

### Step 2: Calculate the Side Length of the Smaller Square
The side length of the smaller square can be calculated as follows:
- The distance between the midpoints of two adjacent sides of the larger square is half the diagonal of the larger square.

The diagonal \(d\) of the larger square can be calculated using the formula:
\[ d = \text{side} \times \sqrt{2} = 20\sqrt{2} \times \sqrt{2} = 40 \]
Thus, the side length \(s\) of the smaller square is:
\[ s = \frac{d}{\sqrt{2}} = \frac{40}{\sqrt{2}} = 20\sqrt{2} \]
However, since we are looking for the midpoints, we need to find the length from the center to a vertex of the smaller square.

### Step 3: Calculate the Length of the Tether
The length of the tether for each goat is half the side length of the smaller square:
\[ \text{Length of tether} = \frac{s}{2} = \frac{20\sqrt{2}}{2} = 10\sqrt{2} \]

### Step 4: Determine the Grazing Area for One Goat
Each goat can graze in a circular area with a radius equal to the length of the tether. The area \(A\) of a circle is given by:
\[ A = \pi r^2 \]
For one goat, the grazing area is:
\[ A = \pi (10\sqrt{2})^2 = \pi (100 \times 2) = 200\pi \]

### Step 5: Calculate the Total Grazing Area for Both Goats
Since both goats are tethered to opposite corners and can graze in their respective circular areas without overlapping, the total grazing area is:
\[ \text{Total Grazing Area} = 2 \times 200\pi = 400\pi \]

### Final Answer
The total grazing area of the two goats is:
\[ \boxed{400\pi} \text{ square meters} \]