Difficulty: Easy
Correct Answer: 421.875 cubic centimetres
Explanation:
Introduction / Context:
This straightforward mensuration problem asks for the volume of a cube. Cubes are special rectangular solids with all sides equal, and their volume formula is one of the most basic in geometry and quantitative aptitude.
Given Data / Assumptions:
Concept / Approach:
The volume V of a cube with side length s is:
V = s^3.
This is because the volume of any rectangular solid is length * breadth * height, and for a cube all three are equal to s. We therefore simply cube the side length.
Step-by-Step Solution:
Step 1: Identify the side length s = 7.5 cm.
Step 2: Use the formula for volume of a cube: V = s^3.
Step 3: Compute s^2 first: 7.5 * 7.5 = 56.25.
Step 4: Now multiply by 7.5 again: 56.25 * 7.5.
Step 5: 56.25 * 7.5 = 421.875.
Step 6: Therefore, V = 421.875 cubic centimetres.
Verification / Alternative check:
We can break the multiplication into smaller steps for checking. 7.5 = 15 / 2, so s^3 = (15 / 2)^3 = 3375 / 8. Dividing 3375 by 8 gives 421.875, which matches the previous calculation exactly. This confirms the correctness of the volume.
Why Other Options Are Wrong:
77.145 cubic centimetres and 39.245 cubic centimetres: These are far too small and result from incorrect multiplication or only squaring instead of cubing.
250 cubic centimetres: This is a rounded guess and not supported by the exact arithmetic; it would correspond to roughly 6.3 cm side, not 7.5 cm.
24.435 cubic centimetres: Extremely small and inconsistent with a side length as large as 7.5 cm.
Common Pitfalls:
A common error is to compute s^2 instead of s^3, forgetting that volume involves all three dimensions. Another mistake is mishandling decimal multiplication, especially when squaring or cubing decimal numbers. Working systematically and, when possible, rewriting decimals as fractions can help avoid arithmetic errors.
Final Answer:
The volume of the cube is 421.875 cubic centimetres.
Discussion & Comments