# How do we calculate the rate of water level increase in a sloped swimming pool?

• Anabelle37
In summary: I would recommend using a forum like physicsforums.com to help solve these types of problems. They have a built in search engine which can help you find the answer you're looking for. Thanks for the input!In summary, the height of the water in a swimming pool is rising at a rate of 1 meter per second as the depth of the water at the deep end is 1 meter.
Anabelle37
URGENT - related rates pool problem

## Homework Statement

A swimming pool with a rectangular surface 18 m long and 12m wide is being filled at the rate of 0.8 cubic metres/min. At one end the pool is 1m deep and at the other end it is 2.5m deep, with a constant slope between the ends. How fast is the height of the water rising when th depth of water at the deep end is 1m?

## The Attempt at a Solution

So i have dV/dt = 0.8 and i know i want to find dh/dt when h=1m from deep end.

dV/dt=dV/dh * dh/dt

The total volume of pool = 12*18*18*.5*1.5*18 = 52488 m^3

I'm having trouble working out the V in terms of h as the depth is different and different ends of the pool. Please help, this is for revision for an exam!

Thanks

Anabelle37 said:

## Homework Statement

A swimming pool with a rectangular surface 18 m long and 12m wide is being filled at the rate of 0.8 cubic metres/min. At one end the pool is 1m deep and at the other end it is 2.5m deep, with a constant slope between the ends. How fast is the height of the water rising when th depth of water at the deep end is 1m?

## The Attempt at a Solution

So i have dV/dt = 0.8 and i know i want to find dh/dt when h=1m from deep end.

dV/dt=dV/dh * dh/dt

The total volume of pool = 12*18*18*.5*1.5*18 = 52488 m^3

I'm having trouble working out the V in terms of h as the depth is different and different ends of the pool. Please help, this is for revision for an exam!

Thanks

Draw a side view of the pool, with the length being 18 m. and the depth being 1 m. at the shallow end and 2.5 m. at the deep end. The lower part of the side view is a right triangle that is 18 m. on one leg and 1.5 m. on the other. By using similar triangles you should be able to get a relationship between the depth of water and the length of the water line. (The width of the pool is always 12 m.)

I would like to ask why your homework questions are URGENT. Is there a good reason, other than procrastination and lack of effort? If so, maybe there is a way to get ahead of the curve on your work.

Thank you Mark.

And turbo, it's urgent as I've tried to answer them and can't and don't want to just look them up online, I want to work them out myself and Physics Forums is good like that as it gives you hints but doesn't solve them for you. And it's for an exam in a few days and the lecturer does not give solutions out!

Anabelle37 said:
Thank you Mark.

And turbo, it's urgent as I've tried to answer them and can't and don't want to just look them up online, I want to work them out myself and Physics Forums is good like that as it gives you hints but doesn't solve them for you. And it's for an exam in a few days and the lecturer does not give solutions out!

## 1. What is a "related rates pool problem"?

A related rates pool problem is a type of mathematics problem that involves finding the rate of change of one quantity with respect to another quantity, where both quantities are changing over time. This type of problem typically involves a scenario where water is being added or drained from a pool, and the goal is to find the rate at which the water level is changing.

## 2. How do I solve a related rates pool problem?

To solve a related rates pool problem, you will need to use the chain rule from calculus. This involves setting up an equation that relates the rates of change of the different quantities in the problem, and then taking the derivative of both sides with respect to time. From there, you can plug in the given values and solve for the desired rate of change.

## 3. What are the key variables in a related rates pool problem?

The key variables in a related rates pool problem are typically the rate of change of the water level in the pool, the rate at which water is being added or drained from the pool, and the dimensions of the pool (such as length, width, and depth). These variables are used to set up the equation and solve for the desired rate of change.

## 4. Can a related rates pool problem have multiple solutions?

Yes, a related rates pool problem can have multiple solutions. It is important to carefully consider the given information and any assumptions made in order to determine if there are multiple valid solutions. Sometimes, there may be more than one way to interpret the problem, leading to different solutions.

## 5. How can related rates pool problems be applied in real life?

Related rates pool problems can be applied in real life in situations where quantities are changing over time, such as filling or draining a pool, changing the water level in a bathtub, or filling a tank with water. This type of problem can also be used in other fields such as physics, engineering, and economics to analyze rates of change in various systems.

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