A rectangular swimming pool of length 20 m and breadth 10 m has a uniformly sloping floor. The depth is 1 m at one end along the length and 3 m at the other end. How much water, in cubic metres, is needed to completely fill this pool?

Introduction / Context:
This problem asks for the volume of water required to fill a swimming pool that has a rectangular top surface but a floor that slopes uniformly from one end to the other. Such pools are common in practical situations, and the question tests understanding of how to treat the varying depth as an average when computing the volume.

Given Data / Assumptions:

• Top surface of the pool is a rectangle of length L = 20 m and breadth B = 10 m.
• Depth at one end along the length is 1 m.
• Depth at the other end is 3 m.
• The floor slopes uniformly, so depth varies linearly from 1 m to 3 m along the length.
• We assume vertical sides and no extra features such as steps that change volume.

Concept / Approach:
For a tank or pool that has a uniform slope in one direction, the depth at any point is a linear function between the shallow and deep ends. The volume can be found by taking the area of the horizontal top surface and multiplying by the average depth. The average depth is simply (shallow depth + deep depth) divided by 2, because of the linear variation.

Step-by-Step Solution:
Step 1: Compute the top surface area of the pool: A = L * B = 20 * 10 = 200 square metres.Step 2: Compute the shallow depth d1 = 1 m.Step 3: Compute the deep depth d2 = 3 m.Step 4: Find the average depth d_avg = (d1 + d2) / 2 = (1 + 3) / 2 = 4 / 2 = 2 m.Step 5: Volume of water required is V = A * d_avg = 200 * 2 = 400 cubic metres.

Verification / Alternative check:
Another way to see this is to imagine the pool volume as the average of two rectangular prisms: one with depth 1 m and one with depth 3 m.Volume with depth 1 m would be 200 * 1 = 200 cubic metres.Volume with depth 3 m would be 200 * 3 = 600 cubic metres.The average of 200 and 600 is (200 + 600) / 2 = 800 / 2 = 400 cubic metres, which matches the earlier computation.

Why Other Options Are Wrong:
An answer of 800 cubic metres corresponds to using the deepest depth 3 m for the entire pool, which overestimates the volume.An answer of 300 or 480 cubic metres arises from incorrect use of the average depth or miscalculations of the area.A value such as 600 cubic metres represents treating the entire pool as if it had the deep depth only.Only 400 cubic metres correctly reflects the average depth of 2 m.

Common Pitfalls:
Learners sometimes simply take the deeper depth and multiply by the area, ignoring the sloping nature of the floor.Another error is to average the lengths instead of the depths, which is not relevant here.Some may attempt unnecessarily complex integration when the linear variation allows a simple average depth method.

Final Answer:
The volume of water needed to fill the pool is 400 cubic metres.