A thin metal iron-sheet is a rhombus in shape, with each side 10 m. If one of its diaginals is 16 m, find the cost of painting its both sides at the rate of Rs. 6 per `m^(2)`.
Also, find the distance between the opposite sides of this rhombus.

Answer

To solve the problem step by step, we will follow the given information and apply the necessary formulas.

### Step 1: Identify the given data
- Each side of the rhombus (s) = 10 m
- One diagonal (d1) = 16 m

### Step 2: Use the property of rhombus diagonals
The diagonals of a rhombus bisect each other at right angles. Let the other diagonal be d2. Since d1 = 16 m, each half of d1 will be:
\[ \frac{d1}{2} = \frac{16}{2} = 8 \text{ m} \]

### Step 3: Apply the Pythagorean theorem
In triangle AOB, where O is the intersection of the diagonals:
- Hypotenuse (s) = 10 m
- One leg (half of d1) = 8 m
- Let the other leg (half of d2) = b

Using the Pythagorean theorem:
\[ s^2 = \left(\frac{d1}{2}\right)^2 + b^2 \]
\[ 10^2 = 8^2 + b^2 \]
\[ 100 = 64 + b^2 \]
\[ b^2 = 100 - 64 \]
\[ b^2 = 36 \]
\[ b = 6 \text{ m} \]

Thus, half of the other diagonal (d2) is 6 m, giving:
\[ d2 = 2b = 2 \times 6 = 12 \text{ m} \]

### Step 4: Calculate the area of the rhombus
The area (A) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d1 \times d2 \]
Substituting the values:
\[ A = \frac{1}{2} \times 16 \times 12 \]
\[ A = \frac{1}{2} \times 192 \]
\[ A = 96 \text{ m}^2 \]

### Step 5: Calculate the cost of painting both sides
The cost of painting is given at Rs. 6 per m². Since we need to paint both sides, we will double the area:
\[ \text{Total area to paint} = 2 \times 96 = 192 \text{ m}^2 \]
Now, calculate the cost:
\[ \text{Cost} = \text{Total area} \times \text{Rate} \]
\[ \text{Cost} = 192 \times 6 \]
\[ \text{Cost} = 1152 \text{ Rs} \]

### Step 6: Find the distance between the opposite sides
The distance (h) between the opposite sides of the rhombus can be found using the area formula:
\[ A = \text{Base} \times \text{Height} \]
Here, the base is the length of one side (10 m):
\[ 96 = 10 \times h \]
\[ h = \frac{96}{10} = 9.6 \text{ m} \]

### Final Answers
- Cost of painting both sides = Rs. 1152
- Distance between opposite sides = 9.6 m