The volume of a closed rectangular metal box with a square base is 4096 `cm^(3)`. The cost of polishing the outer surface of the box is Rs 4 per `cm^(2)`. Find the dimensions of the box for the minimum cost of polishing it.

Answer

To solve the problem of finding the dimensions of a closed rectangular metal box with a square base that minimizes the cost of polishing, we can follow these steps:

### Step 1: Define Variables
Let the side of the square base be \( x \) cm and the height of the box be \( y \) cm.

### Step 2: Write the Volume Equation
The volume \( V \) of the box is given by:
\[ V = x^2 \cdot y \]
Since the volume is given as \( 4096 \, \text{cm}^3 \), we have:
\[ x^2 \cdot y = 4096 \]
From this, we can express \( y \) in terms of \( x \):
\[ y = \frac{4096}{x^2} \]

### Step 3: Write the Surface Area Equation
The surface area \( S \) of the box consists of the area of the two square bases and the four rectangular sides:
\[ S = 2x^2 + 4xy \]
Substituting \( y \) from the previous step:
\[ S = 2x^2 + 4x\left(\frac{4096}{x^2}\right) \]
This simplifies to:
\[ S = 2x^2 + \frac{16384}{x} \]

### Step 4: Write the Cost Function
The cost \( C \) of polishing the surface area at Rs 4 per cm² is given by:
\[ C = 4S = 4\left(2x^2 + \frac{16384}{x}\right) \]
Thus:
\[ C = 8x^2 + \frac{65536}{x} \]

### Step 5: Differentiate the Cost Function
To find the minimum cost, we differentiate \( C \) with respect to \( x \):
\[ \frac{dC}{dx} = 16x - \frac{65536}{x^2} \]

### Step 6: Set the Derivative to Zero
To find the critical points, set the derivative equal to zero:
\[ 16x - \frac{65536}{x^2} = 0 \]
Rearranging gives:
\[ 16x^3 = 65536 \]
Thus:
\[ x^3 = \frac{65536}{16} = 4096 \]
Taking the cube root:
\[ x = 16 \, \text{cm} \]

### Step 7: Find the Height \( y \)
Now substitute \( x = 16 \) back into the equation for \( y \):
\[ y = \frac{4096}{16^2} = \frac{4096}{256} = 16 \, \text{cm} \]

### Conclusion
The dimensions of the box that minimize the cost of polishing are:
\[ \text{Dimensions: } 16 \, \text{cm} \times 16 \, \text{cm} \times 16 \, \text{cm} \]

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