We are asked to find the minimum surface area necessary to construct a rectangular box with a square base and an open top, where the volume is 24 cm³. Let’s define the variables:

• Let $x$ be the side length of the square base (in cm).
• Let $h$ be the height of the box (in cm).
• The volume of the box is given as $V = x^2 h = 24 \, \text{cm}^3$.

Step 1: Express $h$ in terms of $x$

From the volume equation $x^2 h = 24$, we can solve for $h$: $h = \frac{24}{x^2}$

Step 2: Express the surface area in terms of $x$

The surface area of the box consists of:

• The area of the square base: $x^2$,
• The area of the four sides (each side has an area of $x \times h$).

The total surface area $S$ is: $S = x^2 + 4(x \cdot h)$ Substitute $h = \frac{24}{x^2}$ into this equation: $S = x^2 + 4 \left( x \cdot \frac{24}{x^2} \right)$ Simplify: $S = x^2 + \frac{96}{x}$

Step 3: Minimize the surface area

To find the value of $x$ that minimizes the surface area, we first take the derivative of $S$ with respect to $x$ and set it equal to zero.

$\frac{dS}{dx} = 2x - \frac{96}{x^2}$ Set the derivative equal to zero: $2x - \frac{96}{x^2} = 0$ Multiply through by $x^2$ to eliminate the fraction: $2x^3 = 96$ Solve for $x$: $x^3 = 48$ $x = \sqrt[3]{48}$ This simplifies to: $x = 2\sqrt[3]{6}$

Step 4: Find the corresponding height $h$

Now that we know $x = 2\sqrt[3]{6}$, substitute it into the volume equation to find $h$: [ h = \frac{24}{x^2} = \frac{24}{\left(2\sqrt[3]{6}\right)^2} = \frac{24}{4 \cdot \sqrt[3]{36}} ]

Step 5: Calculate the minimum surface area

To finalize, substitute $x = 2\sqrt[3]{6}$ into the surface area formula $S = x^2 + \frac{96}{x}$.

This results in an