You have been asked to design a rectangular box with a square base and an open top. The volume of the box must be 24cm3 . Determine the minimum surface area necessary to construct a box of this volume. Enter an exact answer.

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

You have been asked to design a rectangular box with a square base and an open top. The volume of the box must be 24cm³. Determine the minimum surface area necessary to construct a box of this volume. Enter an exact answer.

Learn how to solve an optimization problem where you design a rectangular box with a square base and an open top. The volume is 24 cm³, and the goal is to find the minimum surface area using calculus and geometry. Ideal for students in Grades 11-12 or early college.

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