Related Rates Problem: Swimming Pool Depth and Filling Rate Calculation

In summary: To summarize, this is a related rates problem involving a swimming pool with varying depth. The pool is being filled at a rate of 1000 gal/min and we need to find the rate at which the depth of water at the deep end is increasing when it reaches 6 feet. The shape of the pool is similar to a triangular block with a cuboidal top. To solve this, we need to find the volume of the block by finding the areas of the triangular and rectangular cross sections.
  • #1
Rizzice
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Related Rates.. again...?

A swimming pool is 50 feet long and 20 feet wide. Its depth varies uniformly from 2 feet at the shallow end to 12 feet at the deep end. (The figure shows a cross-section of the pool.) Suppose that the pool is being filled at the rate of 1000 gal/min. At what rate is the depth of water at the deep end increasing when the depth there is 6 feet? (One gallon of water occupies a volume of approximately 0.1337 cubic feet.)


So first of all, I can't even picture what this is supposed to look like when I draw it. Supposedly this is a related rates problem, but this doesn't go with any of the equations I've used so far (pretty much just volume equation and pythagorean theorem equation).. I don't know what I'm supposed to do, could anyone at least help me get started or walk me through? I'd appreciate it..
 
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  • #2
Just to get you started, the shape of the swimming pool is similar to a wedge or a triangular block with a cuboidal top . How can you find the volume of this block given the dimensions ?
Hint:Find the area of the triangular and rectangular cross sections of the pool.

Arun
 

1. How do you set up a related rates problem involving swimming pool depth and filling rate?

The first step in setting up a related rates problem involving swimming pool depth and filling rate is to identify the variables involved and determine which ones are changing with respect to time. In this case, the variables are the depth of the pool, which is changing, and the filling rate, which is constant. Next, you will need to establish a relationship between the variables, such as the volume formula for a rectangular prism. Finally, you will need to take the derivative with respect to time to create an equation that relates the rates of change.

2. How do you find the rate of change of the pool's depth at a specific time?

To find the rate of change of the pool's depth at a specific time, you will need to differentiate the equation you created in the first step with respect to time. This will give you an equation that relates the rates of change, and you can then plug in the known values for the filling rate and the current depth to solve for the rate of change at that specific time.

3. What is the relationship between the pool's depth and the filling rate?

The relationship between the pool's depth and the filling rate can be described using the volume formula for a rectangular prism, V = lwh. The depth of the pool (h) is one of the dimensions, and the filling rate is equivalent to the rate of change of the volume (dV/dt). Therefore, the relationship can be expressed as dh/dt = (1/A)(dV/dt), where A is the surface area of the pool's base.

4. How can you use related rates to determine the time it takes to fill a pool to a certain depth?

To determine the time it takes to fill a pool to a certain depth using related rates, you will need to solve for time in the equation you created in the first step. This will give you an equation that relates time to the pool's depth and filling rate. You can then plug in the known values for the pool's depth and filling rate to solve for the time it takes to reach the desired depth.

5. What are some real-world applications of related rates problems involving swimming pool depth and filling rate?

Some real-world applications of related rates problems involving swimming pool depth and filling rate include calculating the time it takes to fill a pool to a specific depth, determining the rate at which the pool is filling or draining, and predicting when a pool will be filled to a certain level. These types of problems are also commonly used in engineering and physics to analyze and design systems involving changing volumes and rates of change.

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