A swimming pool $9\text{m}$ wide and $12\text{m}$ long and $1\text{m}$ deep on the shallow side and $4\text{m}$ deep on the deeper side. Its volume is

### Concept & Volume of a Prism A swimming pool with a uniformly sloping bottom can be treated as a prism whose cross-section is a trapezium. The area of a trapezium is given by: $$A = \frac{1}{2} \times (\text{sum of parallel sides}) \times (\text{distance between them})$$ The volume of the pool is the Area of the cross-section multiplied by its width. $$V = A \times w$$ ### Step-by-Step Solution * **Given:** * Length of the pool (distance between parallel sides of trapezium) $= 12 \text{ m}$ * Depths (parallel sides of trapezium) $= 1 \text{ m}$ and $4 \text{ m}$ * Width of the pool $= 9 \text{ m}$ * **Calculation:** 1. Calculate the area of the trapezoidal cross-section (side profile): $$\text{Area} = \frac{1}{2} \times (1 + 4) \times 12$$ $$\text{Area} = \frac{1}{2} \times 5 \times 12 = 5 \times 6 = 30 \text{ m}^2$$ 2. Calculate the total volume by multiplying the cross-sectional area by the width: $$\text{Volume} = 30 \text{ m}^2 \times 9 \text{ m} = 270 \text{ m}^3$$ ### Exam Strategy & Shortcut For any rectangular pool with a uniformly sloping bottom, simply find the "Average Depth". Average Depth $= \frac{\text{Shallow Depth} + \text{Deep Depth}}{2} = \frac{1 + 4}{2} = 2.5 \text{ m}$. Then, calculate volume like a standard cuboid: Volume $= \text{Length} \times \text{Width} \times \text{Average Depth} = 12 \times 9 \times 2.5 = 108 \times 2.5 = 270 \text{ m}^3$. ### Common Pitfall A frequent error is calculating the volume using only one of the depths (yielding $108$ or $432$) or trying to split the shape into a rectangle and a triangle but messing up the dimensions. ### Final Answer Therefore, the correct answer is **$270 \text{ m}^3$**.