A right triangular prism has an equilateral triangle as its base. The side of the equilateral triangular base is 15 centimetres and the height of the prism is 20√3 centimetres. What is the volume of the prism in cubic centimetres?

Introduction / Context:
This question deals with the volume of a right triangular prism whose base is an equilateral triangle. The volume of any prism is the area of its base multiplied by its height. When the base is an equilateral triangle, we use a specific formula for the area of an equilateral triangle in terms of its side length. Such problems are common in solid geometry and engineering applications.

Given Data / Assumptions:
- The base of the prism is an equilateral triangle.
- Side length of the equilateral triangle is 15 cm.
- Height of the prism is 20√3 cm.
- The prism is right, meaning the lateral edges are perpendicular to the base.
- We need to find the volume in cubic centimetres.

Concept / Approach:
Volume of a prism is given by:
Volume = area of base * height
For an equilateral triangle with side a, the area is:
Area = (√3 / 4) * a^2
After computing the area of the triangular base using a = 15 cm, we multiply this area by the prism height 20√3 cm to obtain the volume.

Step-by-Step Solution:
Step 1: Compute the area of the equilateral triangular base using a = 15 cm. Step 2: Area of base = (√3 / 4) * a^2 = (√3 / 4) * 15^2 = (√3 / 4) * 225. Step 3: Simplify this to get area of base = (225√3) / 4 square centimetres. Step 4: Volume of the prism is area of base multiplied by height: Volume = (225√3 / 4) * 20√3. Step 5: Multiply √3 * √3 = 3, so Volume = (225 * 20 * 3) / 4 = (225 * 60) / 4 = 225 * 15 = 3375 cubic centimetres.

Verification / Alternative check:
We can also group the constants differently: (225 / 4) * 20 = 225 * 5 = 1125. Then Volume = 1125 * (√3 * √3) = 1125 * 3 = 3375 cm3. This provides a quick mental cross check and shows that there was no arithmetic slip in the original calculation.

Why Other Options Are Wrong:
1125 cm3 is simply the intermediate product before multiplying by the additional factor of 3 from √3 * √3.
4500 cm3 and 6750 cm3 result from miscalculating the multiplication or missing one of the factors in the formula.
2250 cm3 arises if someone incorrectly uses half the height or misapplies the area formula of the equilateral triangle.

Common Pitfalls:
A common mistake is to forget the √3 factor in the area of an equilateral triangle or to use the formula for the area of a right triangle instead. Another pitfall is to treat the prism as a cylinder or some other solid. Always remember that volume of any prism is the base area times the height and then compute the base area correctly based on the given shape.

Final Answer:
The volume of the right triangular prism is 3375 cm3.