Question

A rectangular swimming pool is 48 m long and 20 m wide. The shallow edge of the pool is 1 m deep. For every 2.6 m that one walks up the inclined base of the swimming pool, one gains an elevation of 1 m. What is the volume of water (in cubic meters), in the swimming pool? Assume that the pool is filled up to the brim.

• A. 528
• B. 960
• C. 6790
• D. 10560
• E. 12960

Hint:

For a pool with a linearly sloping bottom, treat the cross-section as a trapezoid: use $\tan\theta$ to get the depth difference, then multiply footprint area by the {average} of the end depths.

Answer 1

The Correct Option is D

Step 1: Interpret the slope of the base.
“Every $2.6$ m along the inclined base gives $1$ m rise” $⇒ \sin\theta=\dfrac{1}{2.6}=\dfrac{5}{13}$.
Thus $\cos\theta=\dfrac{12}{13}$ and $\tan\theta=\dfrac{\sin\theta}{\cos\theta}=\dfrac{5}{12}$.
Step 2: Depth at the deep end.
Over the horizontal length $L=48$ m, vertical rise \(=\ L\tan\theta=48 \dfrac{5}{12}=20\) m.
Shallow depth is \(1\) m, so deep depth \(=1+20=21\) m.
Step 3: Volume of the pool.
Depth varies linearly from \(1\) to \(21\) along the length, hence average depth \(=\dfrac{1+21}{2}=11\) m.
\[ V=\text{(length)}\times\text{(width)}\times\text{Average depth)} =48\times 20\times 11= \boxed{10560\ \text{m}^3}. \]