The volume of a closed square based rectangular box is 1000 cubic metre. The cost of constructing the base is 15 paise per square metre and the cost of constructing the top is 25 paise per square metre. The cost of constructing its sides is 20 paise per square metre and the cost of constructing the box is Rs. 3.Find the dimensions of box for minimum cost of construction.

Answer

To find the dimensions of a closed square-based rectangular box that minimizes the cost of construction, we can follow these steps:

### Step 1: Define Variables
Let:
- \( L \) = length of the side of the square base (in meters)
- \( h \) = height of the box (in meters)

### Step 2: Volume Constraint
The volume \( V \) of the box is given by:
\[ V = L^2 \cdot h \]
We know that the volume is 1000 cubic meters, so:
\[ L^2 \cdot h = 1000 \]
From this, we can express \( h \) in terms of \( L \):
\[ h = \frac{1000}{L^2} \]

### Step 3: Cost Function
Next, we need to formulate the cost function based on the given costs:
- Cost of constructing the base (15 paise/m²): \( 15 \cdot L^2 \)
- Cost of constructing the top (25 paise/m²): \( 25 \cdot L^2 \)
- Cost of constructing the sides (20 paise/m²): The area of the sides is \( 4 \cdot (L \cdot h) \), so the cost is \( 20 \cdot 4 \cdot (L \cdot h) = 80 \cdot L \cdot h \)

Thus, the total cost \( C \) in paise is:
\[ C = 15L^2 + 25L^2 + 80Lh \]
Combining the base and top costs:
\[ C = 40L^2 + 80Lh \]

### Step 4: Substitute \( h \)
Substituting \( h \) from Step 2 into the cost function:
\[ C = 40L^2 + 80L\left(\frac{1000}{L^2}\right) \]
This simplifies to:
\[ C = 40L^2 + \frac{80000}{L} \]

### Step 5: Differentiate the Cost Function
To find the minimum cost, we differentiate \( C \) with respect to \( L \) and set the derivative equal to zero:
\[ \frac{dC}{dL} = 80L - \frac{80000}{L^2} \]
Setting the derivative to zero:
\[ 80L - \frac{80000}{L^2} = 0 \]

### Step 6: Solve for \( L \)
Rearranging gives:
\[ 80L^3 = 80000 \]
Dividing both sides by 80:
\[ L^3 = 1000 \]
Taking the cube root:
\[ L = 10 \text{ meters} \]

### Step 7: Find \( h \)
Now substituting \( L \) back to find \( h \):
\[ h = \frac{1000}{L^2} = \frac{1000}{10^2} = \frac{1000}{100} = 10 \text{ meters} \]

### Conclusion
The dimensions of the box for minimum cost of construction are:
- Length \( L = 10 \) meters
- Breadth \( B = 10 \) meters (since it is a square base)
- Height \( h = 10 \) meters