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:
$$S=x^2+4\cdot (x\cdot \frac{500}{x^2})$$
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.