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We are tasked with finding the greatest volume of a cuboid whose base is a square and its total surface area equals 150 cm².

### Step 1: Define the variables
- Let the side of the square base be \( x \) cm.
- Let the height of the cuboid be \( h \) cm.

### Step 2: Write the formulas
- The area of the square base is \( x^2 \).
- The total surface area of the cuboid consists of:
  - The area of two square bases: \( 2x^2 \).
  - The area of the four side faces: \( 4xh \) (since there are two faces of height \( h \) and width \( x \)).

Thus, the total surface area \( A \) is:
\[
A = 2x^2 + 4xh
\]
We are given that the total surface area equals 150 cm², so:
\[
2x^2 + 4xh = 150
\]

### Step 3: Solve for \( h \)
Rearrange the surface area formula to solve for \( h \):
\[
4xh = 150 - 2x^2
\]
\[
h = \frac{150 - 2x^2}{4x}
\]

### Step 4: Write the volume formula
The volume \( V \) of the cuboid is the area of the base times the height:
\[
V = x^2 h
\]
Substitute \( h = \frac{150 - 2x^2}{4x} \) into the volume equation:
\[
V = x^2 \left(\frac{150 - 2x^2}{4x}\right)
\]
Simplifying:
\[
V = \frac{x(150 - 2x^2)}{4}
\]
\[
V = \frac{150x - 2x^3}{4}
\]

### Step 5: Maximize the volume
To find the value of \( x \) that maximizes the volume, we take the derivative of \( V \) with respect to \( x \) and set it equal to zero:
\[
\frac{dV}{dx} = \frac{1}{4} \left( 150 - 6x^2 \right) = 0
\]
Solve for \( x \):
\[
150 - 6x^2 = 0
\]
\[
x^2 = 25
\]
\[
x = 5 \, \text{cm}
\]

### Step 6: Find the height and the maximum volume
Substitute \( x = 5 \) cm into the formula for \( h \):
\[
h = \frac{150 - 2(5^2)}{4(5)} = \frac{150 - 50}{20} = \frac{100}{20} = 5 \, \text{cm}
\]

Finally, the volume is:
\[
V = x^2 h = 5^2 \times 5 = 125 \, \text{cm}^3
\]

Thus, the greatest volume of the cuboid is **125 cm³**.

Let me know if you want any further details or have questions. Here are 5 related questions:

1. How do you find the derivative of a function in general?
2. Why is it necessary to maximize the volume using the first derivative?
3. Can this method be applied to find the greatest volume of other shapes?
4. What would happen if the base were not a square?
5. How does the surface area constraint affect the shape of the cuboid?

**Tip:** Always check for critical points by taking the second derivative or analyzing the function to ensure you have a maximum, not a minimum.

Find the greatest volume of the cuboid whose base is a square and its total surface area equals 150 cm².

Find the maximum volume of a cuboid with a square base and a total surface area of 150 cm². Learn how to apply optimization techniques using calculus and geometry. Suitable for students in Grades 10-12.

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