A rhombus-shaped sheet with perimerter 40 cm and on e diagonal 12 cm, is painted on both sides at the rate of Rs. 5 per `cm^(2)`. Find the cost of painting.

Explanation

To find the cost of painting a rhombus-shaped sheet with a perimeter of 40 cm and one diagonal of 12 cm, we can follow these steps:

### Step 1: Find the length of one side of the rhombus.
The perimeter of a rhombus is given by the formula:
\[ \text{Perimeter} = 4 \times \text{side} \]
Given that the perimeter is 40 cm, we can set up the equation:
\[ 4 \times \text{side} = 40 \]
Dividing both sides by 4 gives:
\[ \text{side} = 10 \text{ cm} \]

### Step 2: Find the length of the other diagonal.
In a rhombus, the diagonals bisect each other at right angles. Let the length of the other diagonal be \(d_2\). We know one diagonal \(d_1 = 12 \text{ cm}\). The diagonals divide the rhombus into four right-angled triangles.

Using the Pythagorean theorem in one of the triangles formed by the diagonals, we have:
\[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = \text{side}^2 \]
Substituting the known values:
\[ \left(\frac{12}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = 10^2 \]
This simplifies to:
\[ 6^2 + \left(\frac{d_2}{2}\right)^2 = 100 \]
\[ 36 + \left(\frac{d_2}{2}\right)^2 = 100 \]
Subtracting 36 from both sides:
\[ \left(\frac{d_2}{2}\right)^2 = 64 \]
Taking the square root:
\[ \frac{d_2}{2} = 8 \quad \Rightarrow \quad d_2 = 16 \text{ cm} \]

### Step 3: Calculate the area of the rhombus.
The area \(A\) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
Substituting the values of the diagonals:
\[ A = \frac{1}{2} \times 12 \times 16 \]
Calculating this gives:
\[ A = \frac{1}{2} \times 192 = 96 \text{ cm}^2 \]

### Step 4: Calculate the cost of painting.
Since the sheet is painted on both sides, the total area to be painted is:
\[ \text{Total Area} = 2 \times A = 2 \times 96 = 192 \text{ cm}^2 \]
The cost of painting is given at the rate of Rs. 5 per cm². Therefore, the total cost \(C\) is:
\[ C = 192 \times 5 = 960 \text{ Rs.} \]

### Final Answer:
The cost of painting the rhombus-shaped sheet is Rs. 960.

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