The height of a solid rectangular wall is six times its width and the length of the wall is seven times its height. If the volume of the wall is 16128 cubic metres, find the width of the wall.

Introduction / Context:
This question uses algebra with geometric dimensions. The wall is modelled as a rectangular solid (a cuboid) where the dimensions are linked by simple ratios. By expressing all dimensions in terms of a single variable, we can use the given volume to solve for that variable and hence determine the width of the wall.

Given Data / Assumptions:

• Let width of wall be w metres.
• Height of wall h = 6w.
• Length of wall L = 7h = 7 * 6w = 42w.
• Volume of wall V = 16128 m^3.
• Volume formula for a cuboid: V = L * w * h.

Concept / Approach:
We substitute the expressions for L and h in terms of w into the volume formula. This gives us an equation with a single unknown w. Solving this equation yields w, and then we can check that the resulting height and length make sense with the given volume.

Step-by-Step Solution:
Step 1: Write volume in terms of w. V = L * w * h = (42w) * w * (6w) = 42 * 6 * w^3. Step 2: Compute the constant factor 42 * 6 = 252. So V = 252 * w^3. Step 3: Use the given volume. 252 * w^3 = 16128. Step 4: Solve for w^3. w^3 = 16128 / 252. Step 5: Perform the division. 252 * 64 = 16128, so w^3 = 64. Step 6: Take cube root: w = cube root of 64 = 4 metres.

Verification / Alternative check:
If w = 4 m, then h = 6 * 4 = 24 m and L = 7 * 24 = 168 m. The volume then is L * w * h = 168 * 4 * 24. Multiplying, 168 * 4 = 672, and 672 * 24 = 16128 m^3, which matches the given volume exactly. This confirms that the width of 4 m is correct.

Why Other Options Are Wrong:
If w = 5 m, then w^3 = 125 m^3, and V would be 252 * 125 = 31500 m^3, not 16128 m^3.
For w = 6 m, V = 252 * 216 = 54432 m^3, which is much larger than given.
For w = 7 m or 8 m, the computed volumes are even larger and do not match 16128 m^3.

Common Pitfalls:
A frequent error is misreading the relationships between dimensions, such as confusing length being 7 times the width rather than 7 times the height. Another pitfall is computational accuracy when dividing 16128 by 252 or when finding the cube root. It is helpful to factor numbers or to check small cube values like 4^3 = 64 to quickly identify the correct width.

Final Answer:
The width of the wall is 4 m.