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Related rates of a swimming pool, find rate of change of depth with respect to time.
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Mar4-10, 06:37 AM
1. The problem statement, all variables and given/known dataA swimming pool is 5 m wide, 10 m long, 1 m deep at the shallow end, and 3 m deep at its deepest point. A cross-section is shown in the figure. If the poole is being filled at a rate of 0.1 m^3 per minute, how fast is the water level rising when the depth at the deepest point is 1 m?
2. Relevant equations
None really...but attached is a picture. It is not drawn to scale and all values are given in meters. The blue water given is not given to scale, and is merely used as an aid.
3. The attempt at a solution
Okay, so we know dV/dt = .1 m^3/min, we want dx/dt (let x = water level in the pool (height)), and we need to find some sort of relationship between x and V, and subsequently a relationship between dV/dt and dx/dt.
After much thought, I figured that the volume of the pool will be equal to 50x - 8.5, and i figured that out from:
v = height x length x width minus the small blocks of unused space in the pool
however, that's wrong because when the water level is below 2 m, we're not subtracting some of those blocks...
so i figured that the blocks form triangles, and i could find the rate of change of those triangles' lengths with respect to the height in the pool, x, but i determined that these blocks or triangles aren't remaining as triangles when the pool water level is below 2 meters...they form quadrilaterals...
so i'm stuck, i can't seem to find a relationship between the height of the pool and the volume of the pool...once i find that, it's easy, differentiate implicitly with respect to time, plug in your known values, and then solve for dV/dt...but again, my problem here is finding a relationship between the two.
Any help is greatly appreciated, thanks so much in advance.
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