[Math] FIlling a pool – Related rates

calculusderivatives

Problem

A swimming pool is

  • 25 meters long

  • 10 meters wide

  • 1 meter deep in one end

  • 6 meters deep in the other end

The bottom of the pool is linearly slanted.

The pool is filled with water at a rate of 2000 liters/minute.

How fast does the water level rise at the point in time when the depth of the water is 3 meters?

My work

I have drawn a figure, as I often do.

enter image description here

Since we're dealing with a volume of water, I thought I might start with expressing the volume of the water as a function of the distances of the body of water.

Since the water will be the area of the right triangle, times the width of the pool, I have $$V = \frac12 g h w$$ where $g$ is the horizontal edge of the triangle, $h$ is the current depth of the water, and $w$ is the width of the pool.

Knowing that $h = 3$ and $w = 10$ gives me $V = 15g$.

At this point, I'm stuck, and would appreciate any help!

Best Answer

There are $1000$ liters in a cubic meter, so the fill rate is $2$ m$^3$/min. The slope of the bottom of the pool is $0.2$ (or $-0.2$, depending on your point of view). So when the water is $3$ m deep at the deep end, the horizontal water surface is $15$ m long. Since the pool is $10$ m wide, the surface area at that point is $150$ m.

Instantaneously, then, since you are only interested in first-order changes in water level, you can just pretend the pool has a surface area of $150$ m. How fast does the water level increase if you are pouring water in at $2$ m$^3$/min?

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