The volume of a cube is 166.375 cm³. What is the length (in cm) of each side of the cube?

Introduction / Context:
This mensuration question asks you to find the side length of a cube when its volume is given. Since all edges of a cube are equal, the volume formula is simple and involves the cube of the side length. This is a common type of problem in aptitude tests and basic geometry.

Given Data / Assumptions:

• The solid is a cube.
• Volume of the cube is 166.375 cm³.
• Let the side length be s cm.
• The volume of a cube is given by V = s³.
• We must find s.

Concept / Approach:
We use the direct relationship between volume and side length for a cube: V = s³ So the side length s is the cube root of the volume: s = ∛V We either compute the cube root directly or test the provided options by cubing them to see which one gives the exact volume 166.375 cm³. Since the options are decimals with one decimal place, checking them is straightforward.

Step-by-Step Solution:
Step 1: Use the formula V = s³, where V = 166.375 cm³. Step 2: We need s such that s³ = 166.375. Step 3: Try s = 5.5 cm from the options. Step 4: Compute 5.5² = 30.25. Step 5: Then 5.5³ = 5.5 * 30.25 = 166.375. Step 6: Since this exactly matches the given volume, the side length s is 5.5 cm.

Verification / Alternative check:
We can quickly verify that no other option gives the correct volume. For example, 4.5³ = 91.125 cm³ (too small), 6.5³ = 274.625 cm³ (too large), 7.5³ = 421.875 cm³ (much too large) and 8.5³ = 614.125 cm³. Only 5.5³ equals 166.375 cm³. This confirms that 5.5 cm is the unique side length that matches the given volume.

Why Other Options Are Wrong:
Any side length smaller than 5.5 cm (such as 4.5 cm) produces a significantly smaller volume than 166.375 cm³ because volume grows rapidly with the cube of side length. Any side length larger than 5.5 cm (such as 6.5, 7.5 or 8.5 cm) produces much larger volumes. None of these equal the target volume, so they cannot be correct side lengths for this cube.

Common Pitfalls:
A common error is to mistakenly square the side instead of cubing it or to take the square root instead of the cube root when working backwards. Another pitfall is rounding intermediate steps too aggressively, which can hide the exact equality between a candidate side length and the given volume. When options are given, directly cubing each option is often the quickest and most reliable way to find the correct answer.

Final Answer:
The side length of the cube is 5.5 cm.