The cross section of a canal is in the shape of an isosceles trapezium that is 3 m wide at the bottom and 5 m wide at the top. If the depth of the canal is 2 m and its length is 110 m, what is the maximum capacity of the canal, in cubic metres, when it is completely filled with water?

Introduction / Context:
This question combines the area of a trapezium with the concept of volume. The cross section of the canal is a trapezium, and by finding its area and multiplying by the length of the canal, we determine the total volume it can hold. This type of geometry problem is very common in civil engineering and competitive exams related to mensuration.

Given Data / Assumptions:

• Cross section is an isosceles trapezium.

• Bottom width (one parallel side) = 3 m.

• Top width (other parallel side) = 5 m.

• Depth (distance between parallel sides) = 2 m.

• Length of the canal = 110 m.

• Water fills the entire cross section along the full length.

Concept / Approach:
First, compute the area of the trapezium that forms the cross section. The area of a trapezium with parallel sides a and b and distance between them h is given by (1 / 2) * (a + b) * h. Here, the parallel sides are the bottom and top widths, and the distance between them is the depth. Once the area of the cross section is known, multiply it by the length of the canal to get the total volume or capacity.

Step-by-Step Solution:
Let a = 3 m (bottom width), b = 5 m (top width), h = 2 m (depth). Area of trapezium = (1 / 2) * (a + b) * h. Substitute values: Area = (1 / 2) * (3 + 5) * 2. Compute inside bracket: 3 + 5 = 8. Area = (1 / 2) * 8 * 2. Area = (1 / 2) * 16 = 8 square metres. Length of canal L = 110 m. Volume (capacity) = cross sectional area * length = 8 * 110. Volume = 880 cubic metres.

Verification / Alternative check:
We can also reason that the average width of the canal is the mean of the top and bottom widths, that is (3 + 5) / 2 = 4 m. If we treat the cross section as equivalent to a rectangle of width 4 m and height 2 m, then its area is 4 * 2 = 8 square metres, which matches the trapezium area calculation. Multiplying by the length 110 m again yields 8 * 110 = 880 cubic metres, confirming the result.

Why Other Options Are Wrong:
The value 1760 cubic metres is exactly twice the correct capacity and would come from mistakenly doubling the cross sectional area. The value 1650 cubic metres might arise from mis adding widths or using 7 as the average width instead of 4. The value 1056 cubic metres suggests a mixed calculation error, perhaps by taking an incorrect height or length. Only 880 cubic metres matches both the trapezium area and the volume formula correctly.

Common Pitfalls:
Students may mistakenly use only one width instead of the average of the two parallel sides, effectively treating the shape as a rectangle instead of a trapezium. Another common error is to mix up the depth and length or forget to multiply by the full canal length. Carefully distinguishing between cross sectional dimensions and longitudinal dimensions is crucial when moving from area to volume.

Final Answer:
The maximum capacity of the canal is 880 cubic metres.