An open box is made of wood 2 cm thick. Its internal length is 86 cm , breadth 46 cm and height is 38 cm. The cost of painting the outer surface of the box at Rs. 10 per `m^(2)` is :

Options

A. Rs. 18.5
B. Rs. 8.65
C. Rs. 11.65
D. `Rs. 17.50`

Answer

To find the cost of painting the outer surface of the open box, we will follow these steps:

### Step 1: Calculate the outer dimensions of the box.
The box is made of wood that is 2 cm thick. Therefore, we need to add 2 cm to each side of the internal dimensions to find the outer dimensions.

- **Internal Length** = 86 cm
- **Internal Breadth** = 46 cm
- **Internal Height** = 38 cm

**Outer Length** = Internal Length + 2 cm (thickness) + 2 cm (thickness)
= 86 cm + 2 cm + 2 cm
= 90 cm

**Outer Breadth** = Internal Breadth + 2 cm (thickness) + 2 cm (thickness)
= 46 cm + 2 cm + 2 cm
= 50 cm

**Outer Height** = Internal Height + 2 cm (thickness) + 2 cm (thickness)
= 38 cm + 2 cm + 2 cm
= 42 cm

### Step 2: Convert the outer dimensions from centimeters to meters.
Since the cost of painting is given per square meter, we need to convert the dimensions into meters.

- **Outer Length** = 90 cm = 90/100 m = 0.9 m
- **Outer Breadth** = 50 cm = 50/100 m = 0.5 m
- **Outer Height** = 42 cm = 42/100 m = 0.42 m

### Step 3: Calculate the outer surface area of the open box.
The formula for the surface area of an open box (without a top) is:
\[ \text{Surface Area} = \text{Length} \times \text{Breadth} + 2 \times (\text{Length} \times \text{Height}) + 2 \times (\text{Breadth} \times \text{Height}) \]

Substituting the outer dimensions:
\[ \text{Surface Area} = (0.9 \times 0.5) + 2 \times (0.9 \times 0.42) + 2 \times (0.5 \times 0.42) \]

Calculating each term:
1. \( 0.9 \times 0.5 = 0.45 \)
2. \( 2 \times (0.9 \times 0.42) = 2 \times 0.378 = 0.756 \)
3. \( 2 \times (0.5 \times 0.42) = 2 \times 0.21 = 0.42 \)

Adding these together:
\[ \text{Surface Area} = 0.45 + 0.756 + 0.42 = 1.626 \, m^2 \]

### Step 4: Calculate the cost of painting the outer surface.
The cost of painting is given as Rs. 10 per square meter. Therefore, the total cost can be calculated as:
\[ \text{Total Cost} = \text{Surface Area} \times \text{Cost per m}^2 \]
\[ \text{Total Cost} = 1.626 \times 10 = Rs. 16.26 \]

### Final Answer:
The cost of painting the outer surface of the box is **Rs. 16.26**.

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