A rhombus -shaped field has green grass for `18` cows to graze. If each side of the rhombus is `"30 m"` and its longer diagonal is `"48 m",` how much area of grass field will each cow be getting ?

Options

A. `46m^(2)`
B. `48m^(2)`
C. `45m^(2)`
D. `50m^(2)`

Explanation

To find the area of the rhombus-shaped field and determine how much area each cow will get, we can follow these steps:

### Step 1: Calculate the area of the rhombus using the diagonals.
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 \( d_1 = 48 \, m \) (the longer diagonal). To find the other diagonal \( d_2 \), we can use the properties of the rhombus.

### Step 2: Calculate the length of the second diagonal.
In a rhombus, the diagonals bisect each other at right angles. Let \( d_2 \) be the length of the shorter diagonal. We can use the Pythagorean theorem in one of the right triangles formed by the diagonals.

Each side of the rhombus is \( 30 \, m \), and half of the longer diagonal is \( \frac{48}{2} = 24 \, m \). Let \( \frac{d_2}{2} \) be the half-length of the shorter diagonal. According to the Pythagorean theorem:
\[ \left(\frac{d_2}{2}\right)^2 + 24^2 = 30^2 \]
\[ \left(\frac{d_2}{2}\right)^2 + 576 = 900 \]
\[ \left(\frac{d_2}{2}\right)^2 = 900 - 576 = 324 \]
\[ \frac{d_2}{2} = \sqrt{324} = 18 \]
Thus, \( d_2 = 36 \, m \).

### Step 3: Calculate the area of the rhombus.
Now that we have both diagonals, we can substitute them into the area formula:
\[ A = \frac{1}{2} \times 48 \times 36 \]
\[ A = \frac{1}{2} \times 1728 = 864 \, m^2 \]

### Step 4: Calculate the area available for each cow.
To find the area available for each cow, we divide the total area by the number of cows:
\[ \text{Area per cow} = \frac{864}{18} = 48 \, m^2 \]

### Final Answer:
Each cow will get an area of \( 48 \, m^2 \).

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