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

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To solve the problem of finding the dimensions of a box with a square base and an open top that maximizes volume given a surface area of 192 sq. cm, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( x \) = length of one side 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 with an open top can be expressed as:
\[ S = x^2 + 4xh \]
Given that the surface area is 192 sq. cm, we have:
\[ x^2 + 4xh = 192 \]

### Step 3: Solve for Height \( h \)
Rearranging the surface area equation to 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 can be expressed as:
\[ V = x^2h \]
Substituting \( h \) from the previous step:
\[ V = x^2 \left( \frac{192 - x^2}{4x} \right) \]
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 \( h \)
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: State the Dimensions
The dimensions of the box that maximize the volume are:
- Base side length \( x = 8 \) cm
- Height \( h = 4 \) cm

### Summary of the Solution
The dimensions of the box that maximize the volume while maintaining a surface area of 192 sq. cm are:
- Length of the base: 8 cm
- Height: 4 cm