https://file.aiquickdraw.com/mathbot/file/17329132332598r7gal8q.jpg

To solve the problem, let's analyze it step by step.

The problem describes a rectangular prism with a square base and requires us to maximize its volume, given the total sum of all edges is **504 cm**.

### Step 1: Define Variables
Let:
- \( a \) = side length of the square base
- \( h \) = height of the rectangular prism

Since the base is square:
- The total sum of edges is:
  \[
  4a + 4h = 504
  \]
  Simplifying:
  \[
  a + h = 126 \tag{1}
  \]

### Step 2: Volume of the Prism
The volume \( V \) of the prism is:
\[
V = a^2 \cdot h \tag{2}
\]

### Step 3: Substitute \( h \) from Equation (1)
From \( a + h = 126 \):
\[
h = 126 - a \tag{3}
\]

Substitute \( h \) into the volume formula:
\[
V = a^2 (126 - a)
\]
\[
V = 126a^2 - a^3 \tag{4}
\]

### Step 4: Maximize Volume
To find the value of \( a \) that maximizes \( V \), take the derivative of \( V \) with respect to \( a \) and set it to zero:
\[
\frac{dV}{da} = 252a - 3a^2
\]
\[
252a - 3a^2 = 0
\]
Factorize:
\[
a(252 - 3a) = 0
\]
\[
a = 0 \quad \text{or} \quad a = 84
\]

Since \( a = 0 \) is not feasible, we have:
\[
a = 84
\]

### Step 5: Calculate \( h \)
Substitute \( a = 84 \) into Equation (3):
\[
h = 126 - 84 = 42
\]

### Step 6: Verify Dimensions
The dimensions of the box are:
\[
\text{Base side length } a = 84 \, \text{cm}, \, \text{Height } h = 42 \, \text{cm}.
\]
Thus, the total edges are:
\[
4a + 4h = 4(84) + 4(42) = 336 + 168 = 504 \, \text{cm},
\]
which satisfies the problem's constraint.

### Answer:
The dimensions that maximize the volume are:
\[
\boxed{42 \, \text{cm}, \, 42 \, \text{cm}, \, \text{and } 42 \, \text{cm} \, \text{(Option D)}}
\]

Would you like a detailed explanation or have further questions?  
Here are 5 related questions to explore further:
1. How does the derivative confirm maximum volume in optimization problems?
2. What if the base were not square—how would the calculations change?
3. Can you explain why \( a = 0 \) was not valid for this problem?
4. How does the constraint \( 4a + 4h = 504 \) shape the solution space?
5. What are other methods to optimize volume in rectangular prisms?

**Tip:** Always verify

A rectangular-prism shaped box has a square base. If the sum of all its edges equals 504 cm, calculate the dimensions of the box that will maximize its volume.

Learn how to calculate the dimensions of a rectangular prism with a square base to maximize its volume when the total sum of edges equals 504 cm. Explore key concepts in optimization, volume formulas, and derivative techniques suitable for Grades 11-12.

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