Find the dimensions of a rectangular box with the largest volume if the total surface area is 216 cm^2
. Enter your answers without including units. 

Length = ?

Width = ?

Height = ?

We are tasked with finding the dimensions of a rectangular box with the largest volume given that the total surface area is 216 cm².

### Step 1: Define variables
Let the dimensions of the rectangular box be:
- Length = \( l \)
- Width = \( w \)
- Height = \( h \)

The volume of the rectangular box is:
\[
V = l \cdot w \cdot h
\]
The surface area of the rectangular box is:
\[
S = 2lw + 2lh + 2wh = 216
\]
We want to maximize the volume \( V = l \cdot w \cdot h \) subject to the constraint \( 2lw + 2lh + 2wh = 216 \).

### Step 2: Express one variable in terms of others
To simplify, let's assume the box is a **cube**, i.e., \( l = w = h \), which is typical for maximizing volume in problems with fixed surface area. Then, the surface area equation becomes:
\[
2l^2 + 2l^2 + 2l^2 = 216 \quad \Rightarrow \quad 6l^2 = 216
\]
Solving for \( l \):
\[
l^2 = \frac{216}{6} = 36 \quad \Rightarrow \quad l = \sqrt{36} = 6
\]
Since \( l = w = h \), we conclude that:
\[
\text{Length} = 6, \quad \text{Width} = 6, \quad \text{Height} = 6
\]

### Final Answer:
- Length = 6
- Width = 6
- Height = 6

Would you like more details or have any questions?

Here are 5 related questions to expand the topic:
1. What is the volume of the box with the given dimensions?
2. How would the solution change if the surface area were 432 cm²?
3. What would happen to the volume if the box were not a cube?
4. How can you find the maximum volume if only two sides are equal?
5. How does surface area affect the volume of geometric shapes?

**Tip:** In problems involving maximum volume with constraints, checking symmetric shapes like cubes often simplifies the problem!

Find the dimensions of a rectangular box with the largest volume if the total surface area is 216 cm^2. Enter your answers without including units. Length = ? Width = ? Height = ?

Discover how to find the dimensions of a rectangular box that maximizes volume with a fixed surface area of 216 cm². Learn the step-by-step process using geometric formulas for surface area and volume. Perfect for students in Grades 9-12.

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