A box with a square base is to have an open top. The surface area of the box is 192sq.cm. What should be its dimensions in order that the volume is largest?

Answer

To solve the problem of maximizing the volume of a box with a square base and an open top, given a surface area of 192 cm², we can follow these steps:

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

### Step 2: Write the Surface Area Equation
The surface area \( S \) of the box can be expressed as:
\[ S = x^2 + 4xh \]
Given that the surface area is 192 cm², we have:
\[ x^2 + 4xh = 192 \]

### Step 3: Express Height in Terms of Base Length
From the surface area equation, we can express \( h \) in terms of \( x \):
\[ 4xh = 192 - x^2 \]
\[ h = \frac{192 - x^2}{4x} \]

### Step 4: Write the Volume Equation
The volume \( V \) of the box is given by:
\[ V = x^2h \]
Substituting the expression for \( h \):
\[ V = x^2 \left(\frac{192 - x^2}{4x}\right) \]
This simplifies to:
\[ V = \frac{x(192 - x^2)}{4} \]
\[ V = \frac{192x - x^3}{4} \]

### Step 5: Differentiate the Volume Function
To find the maximum volume, we differentiate \( V \) with respect to \( x \):
\[ \frac{dV}{dx} = \frac{1}{4}(192 - 3x^2) \]

### Step 6: Set the Derivative to Zero
Setting the derivative equal to zero to find critical points:
\[ 192 - 3x^2 = 0 \]
\[ 3x^2 = 192 \]
\[ x^2 = 64 \]
\[ x = 8 \quad (\text{since } x \text{ must be positive}) \]

### Step 7: Find the Height
Now, substitute \( x = 8 \) back into the equation for \( h \):
\[ h = \frac{192 - 8^2}{4 \cdot 8} \]
\[ h = \frac{192 - 64}{32} \]
\[ h = \frac{128}{32} = 4 \]

### Step 8: Conclusion
The dimensions of the box that maximize the volume are:
- Side length of the square base \( x = 8 \) cm
- Height \( h = 4 \) cm

### Summary
The box with the largest volume, given a surface area of 192 cm², has dimensions:
- Base side length: 8 cm
- Height: 4 cm

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