The base area of a right pyramid is 57 square units and its vertical height is 10 units. What is the volume (in cubic units) of the pyramid?

Introduction / Context:
This mensuration problem deals with the volume of a right pyramid. A right pyramid has its apex directly above the centroid of the base, and the standard volume formula applies directly using the base area and the vertical height. It is similar in spirit to the volume formula for cones and prisms but with a different constant factor.

Given Data / Assumptions:

• The solid is a right pyramid.
• Base area A = 57 square units.
• Vertical height h = 10 units.
• We are asked to find the volume in cubic units.
• The volume formula for any pyramid is V = (1 / 3) * base area * height.

Concept / Approach:
The volume of a pyramid is one third of the volume of a prism that has the same base area and height. This leads to the simple formula: V = (1 / 3) * A * h We are given A and h directly, so we only need to substitute and simplify the numeric expression. There is no need to know the exact shape of the base as long as its area is known.

Step-by-Step Solution:
Step 1: Recall the volume formula: V = (1 / 3) * base area * height. Step 2: Substitute the given base area A = 57 square units and height h = 10 units. Step 3: V = (1 / 3) * 57 * 10. Step 4: Compute 57 * 10 = 570. Step 5: Divide by 3: V = 570 / 3 = 190 cubic units. Step 6: Therefore, the volume of the pyramid is 190 cubic units.

Verification / Alternative check:
We can think of constructing a prism with the same base area 57 and height 10, which would have volume 57 * 10 = 570 cubic units. A pyramid with the same base and height fits three times inside such a prism (assuming the appropriate arrangement), which is why the formula has the factor 1 / 3. Thus, 570 divided by 3 naturally gives 190 cubic units. This reasoning confirms the formula and the numerical answer.

Why Other Options Are Wrong:
380 cubic units and 570 cubic units correspond to forgetting the 1 / 3 factor or using 2 / 3 instead. 540 cubic units and 600 cubic units are significantly larger than the prism volume of 570 cubic units when using the given base and height, so they cannot possibly be the volume of a pyramid whose volume is strictly less than that of the corresponding prism. Only 190 cubic units fits the correct formula and logical bounds for the volume.

Common Pitfalls:
A common mistake is to use the prism formula V = A * h and forget the 1 / 3 factor that applies specifically to pyramids. Another pitfall is confusion between slant height and vertical height; here, the problem explicitly gives the vertical height, which is what the formula requires. Carefully reading the wording and recalling that pyramids have one third the volume of a corresponding prism helps avoid these errors.

Final Answer:
The volume of the pyramid is 190 cubic units.