A closed rectangular box has length = 40 cm, breadth = 30 cm and height = 50 cm. It is made of thin metal sheet. Find the cost of metal sheet required to make 20 such boxes, if `1 m^(2)` of metal sheet costs Rs.45.

Answer

To find the cost of the metal sheet required to make 20 closed rectangular boxes with given dimensions, we will follow these steps:

### Step 1: Calculate the Surface Area of One Box
The formula for the surface area \( A \) of a closed rectangular box (cuboid) is given by:
\[ A = 2 \times (l \times b + b \times h + h \times l) \]
where \( l \) is the length, \( b \) is the breadth, and \( h \) is the height.

Given:
- Length \( l = 40 \) cm
- Breadth \( b = 30 \) cm
- Height \( h = 50 \) cm

Substituting the values:
\[ A = 2 \times (40 \times 30 + 30 \times 50 + 50 \times 40) \]

### Step 2: Calculate Each Area Component
Calculating each term:
- \( 40 \times 30 = 1200 \) cm²
- \( 30 \times 50 = 1500 \) cm²
- \( 50 \times 40 = 2000 \) cm²

Now, sum these areas:
\[ A = 2 \times (1200 + 1500 + 2000) \]
\[ A = 2 \times 4700 = 9400 \text{ cm}^2 \]

### Step 3: Calculate the Total Surface Area for 20 Boxes
To find the total surface area for 20 boxes:
\[ \text{Total Surface Area} = 20 \times 9400 = 188000 \text{ cm}^2 \]

### Step 4: Convert Surface Area to m²
Since the cost is given per square meter, we need to convert the area from cm² to m²:
\[ 1 \text{ m}^2 = 10000 \text{ cm}^2 \]
\[ \text{Total Surface Area in m}^2 = \frac{188000}{10000} = 18.8 \text{ m}^2 \]

### Step 5: Calculate the Cost of the Metal Sheet
The cost of 1 m² of metal sheet is Rs. 45. Therefore, the total cost for 18.8 m² is:
\[ \text{Total Cost} = 18.8 \times 45 = 846 \text{ Rs.} \]

### Final Answer
The cost of the metal sheet required to make 20 boxes is Rs. 846.

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