Solving for Rate of Change of Depth in Swimming Pool

• stripes
In summary, the volume of the pool is given by V = 50x - 8.5(2-x)^2 and the rate of change of the depth is given by dx/dt = 0.1/(50 - 17(2-x)). When the deepest point of the pool is 1 m, the water level is rising at a rate of 0.0024 m/min.
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Homework Statement

A 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?

Homework 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.

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.

Attachments

• pool.jpg
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The volume of the pool is given by V = 50x - 8.5(2-x)^2, where x is the height of the pool. To find the rate of change of the depth, we need to differentiate with respect to time. So, dV/dt = 50dx/dt - (8.5)(2-x)d(2-x)/dtIf we assume that the water is filling the pool at a constant rate of 0.1 m^3 per minute, then dV/dt = 0.1. We can substitute this into our equation and solve for dx/dt:0.1 = 50dx/dt - (17)(2-x)dx/dtdx/dt = 0.1/(50 - 17(2-x))When the deepest point of the pool is 1 m, we have x = 2. Substituting this into our equation gives:dx/dt = 0.1/(50 - 17) = 0.0024 m/minTherefore, when the deepest point of the pool is 1 m, the water level is rising at a rate of 0.0024 m/min.

1. What is the rate of change of depth in a swimming pool?

The rate of change of depth in a swimming pool refers to how quickly the depth of the water is changing over a specific period of time. It can be calculated by dividing the change in depth by the change in time.

2. Why is it important to calculate the rate of change of depth in a swimming pool?

Calculating the rate of change of depth in a swimming pool is important for maintaining the proper water level for safety and comfort. It can also help identify any potential leaks or issues with the pool's drainage system.

3. What factors can affect the rate of change of depth in a swimming pool?

The rate of change of depth in a swimming pool can be affected by several factors, such as the size and shape of the pool, the water temperature, the type of filter system used, and any external factors like rainfall or evaporation.

4. How can the rate of change of depth in a swimming pool be measured?

The rate of change of depth in a swimming pool can be measured using a ruler or measuring tape to determine the change in depth, and a stopwatch or timer to measure the change in time. This can be done manually or with the help of specialized tools like flow meters.

5. How can the rate of change of depth in a swimming pool be controlled?

The rate of change of depth in a swimming pool can be controlled by adjusting the pool's drainage system, adding or removing water from the pool, or using a cover to reduce evaporation. Regular maintenance and monitoring can also help keep the rate of change of depth under control.

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