Explanation:
To find the dimensions of a rectangular box with a square base and no top that requires the least amount of material, follow these steps:

1. Define the Variables:
- Let \( x \) be the length of each side of the square base.
- Let \( h \) be the height of the box.

2. Volume Constraint:
The volume \( V \) of the box is given by:
\[
V = x^2 \cdot h = 500 \text{ cm}^3
\]
Thus:
\[
h = \frac{500}{x^2}
\]

3. Surface Area Calculation:
The surface area \( S \) of the box with no top is:
\[
S = x^2 + 4 \cdot (x \cdot h)
\]
Substitute \( h \) from the volume constraint:
\[
S = x^2 + 4 \cdot \left(x \cdot \frac{500}{x^2}\right)
\]
Simplify:
\[
S = x^2 + \frac{2000}{x}
\]

4. Minimize the Surface Area:
To find the value of \( x \) that minimizes \( S \), take the derivative of \( S \) with respect to \( x \) and set it to zero:
\[
\frac{dS}{dx} = 2x - \frac{2000}{x^2}
\]
Set the derivative to zero:
\[
2x - \frac{2000}{x^2} = 0
\]
\[
2x^3 = 2000
\]
\[
x^3 = 1000
\]
\[
x = \sqrt[3]{1000} = 10
\]

5. Calculate \( h \):
Substitute \( x = 10 \) into the volume equation:
\[
h = \frac{500}{10^2} = \frac{500}{100} = 5
\]

6. Dimensions:
The dimensions of the box that minimize the surface area are:
\[
10 \text{ cm} \times 10 \text{ cm} \text{ base} \times 5 \text{ cm} \text{ height}
\]


The dimensions of the box that require the least amount of material are 10 cm x 10 cm x 5 cm.

The correct answer is A. 10 x 10 x 5 cm.