Introduction / Context:
Here we have a cube whose total surface area is given and we are required to find its volume. This connects two standard formulas of a cube: the formula for surface area and the formula for volume. The problem tests the ability to move from area to side length and then from side length to volume, which is a common sequence in aptitude questions on three dimensional geometry.
Given Data / Assumptions:
- Total surface area of the cube, S = 1734 square centimetres.
- Let side of the cube be a centimetres.
- Cube is solid and all edges are equal.
Concept / Approach:
For a cube with side length a:
- Total surface area S = 6 * a^2.
- Volume V = a^3.
We are given S and need V. So we first solve 6 * a^2 = 1734 to find a, and then compute a^3.
Step-by-Step Solution:
Step 1: Use surface area formula.
S = 6 * a^2 = 1734.
Step 2: Solve for a^2.
a^2 = 1734 / 6.
a^2 = 289.
Step 3: Take square root.
a = square root of 289 = 17 centimetres.
Step 4: Compute volume of the cube.
V = a^3 = 17^3.
V = 17 * 17 * 17 = 289 * 17 = 4913 cubic centimetres.
Thus, the volume of the cube is 4913 cubic centimetres.
Verification / Alternative check:
We can verify by reversing the steps. If side length a = 17 cm, then surface area S = 6 * 17^2 = 6 * 289 = 1734 square centimetres, which matches the given value. This confirms that a has been calculated correctly and thus the volume is correct as well.
Why Other Options Are Wrong:
Option A (2334 cubic cm) and Option B (3356 cubic cm) do not correspond to any cube with an integer side that satisfies the given surface area. Option D (3478 cubic cm) is not equal to 17^3 and is not consistent with the surface area. Option E (4096 cubic cm) is 16^3, which would imply a cube with side 16 cm and surface area 6 * 256 = 1536 square centimetres, not 1734.
Common Pitfalls:
A common error is to forget that the surface area formula includes a factor of 6 and to equate a^2 directly with 1734, which is incorrect. Others may compute a^2 correctly but make arithmetic mistakes when cubing 17. It is also important to keep units consistent and recognize that we are moving from area units to volume units, which explains the change from squared to cubed centimetres.
Final Answer:
The volume of the cube is
4913 cubic centimetres.
Discussion & Comments