A rhombus shaped field has green grass for 16 cows to graze. If each side of the rhombus is 40 m and its diagonal is 48 m, then the area of the grass field which each cow can graze is

Options

A. `72m^(2)`
B. `48 m^(2)`
C. `96m^(2)`
D. `128m^(2)`

Answer

To find the area of the rhombus-shaped field that each cow can graze, we will follow these steps:

### Step 1: Calculate the area of the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals. We know one diagonal \( d_1 = 48 \) m. To find the other diagonal \( d_2 \), we can use the properties of the rhombus.

### Step 2: Find the length of the second diagonal
In a rhombus, the diagonals bisect each other at right angles. We can use the Pythagorean theorem to find the second diagonal \( d_2 \). Each side of the rhombus is 40 m, and half of the first diagonal is \( \frac{48}{2} = 24 \) m.

Let \( \frac{d_2}{2} \) be the half-length of the second diagonal. According to the Pythagorean theorem:
\[ \left(\frac{d_2}{2}\right)^2 + 24^2 = 40^2 \]
\[ \left(\frac{d_2}{2}\right)^2 + 576 = 1600 \]
\[ \left(\frac{d_2}{2}\right)^2 = 1600 - 576 = 1024 \]
Taking the square root:
\[ \frac{d_2}{2} = 32 \implies d_2 = 64 \text{ m} \]

### Step 3: Calculate the area of the rhombus using both diagonals
Now that we have both diagonals:
- \( d_1 = 48 \) m
- \( d_2 = 64 \) m

We can substitute these values into the area formula:
\[ A = \frac{1}{2} \times 48 \times 64 \]
\[ A = \frac{1}{2} \times 3072 = 1536 \text{ m}^2 \]

### Step 4: Calculate the area available for each cow
Since the area of the rhombus is 1536 m² and there are 16 cows, the area available for each cow is:
\[ \text{Area per cow} = \frac{1536}{16} = 96 \text{ m}^2 \]

### Final Answer
The area of the grass field which each cow can graze is \( \boxed{96} \text{ m}^2 \).

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