# Calculus application (rate of water draining)

• pezlet
In summary, the conversation discusses a problem of finding the rate of liquid flow from a cylindrical container with known data of height, diameter, and volume. The person is seeking help in setting up the problem and mentions using the Bernoulli's equation as a potential solution. They also mention taking measurements to retry the problem if necessary.

## Homework Statement

This is actually just a problem that I'm trying to work out on my own (working on a little project).
The ultimate goal is that I'll know the amount of liquid coming out of the spigot per second, so I'll know roughly how long to open the spigot for to fill up a container of known volume.

I'm trying to find the rate of a liquid coming out of a spigot from a cylindrical container.
The known data will be:
height of container = 25cm
diameter of container = 12cm

density of liquid (if I need it)
height of liquid (if I need it)

## Homework Equations

volume of the container = 3600π cm^3 or 11310cm^3

## The Attempt at a Solution

I'm honestly pretty stumped from the start. I was thinking of a simple "draining tank problem" like we all go through in our first calculus course, however, I know neither dh/dt or dv/dt.

I can take some empirical data if I need to, but even then I'm not sure where to start.
I could open the spigot for 60 seconds and see how much liquid drained, but that rate will obviously change when there's less water.

I figured that since the spigot is at the bottom of the container, the force pushing down the water through the spigot is gravity acting on the mass of the overall liquid in the container at any given point. I can find the density of the liquid, because I think I will need that to solve the problem.

I honestly need more help setting up the problem than solving it. I've aced all of my calculus and my differential equations classes, but that was a couple years ago and the problem was always more obvious.

Any thoughts to point me in the right direction? Am I missing some desperately needed variables? If I need to take some measurements and retry the problem I absolutely can.

You can use the Bernoulli's equation. With some simplifying assumptions, the data you have will be enough to determine the rate of flow.

Start from the Bernoulli's equation and see how it looks for your system.

That's a really good idea, I had heard of that and couldn't think of what it was called.

Thank you so much, that will definitely get me started.

## 1. How is calculus used to determine the rate of water draining?

Calculus is used to determine the rate of water draining by analyzing the relationship between the volume of water in a container and the time it takes for the water to drain. By taking the derivative of the volume function with respect to time, we can find the instantaneous rate of change, or the rate at which the water is draining at any given moment.

## 2. Can calculus be used to predict the time it will take for a container to drain completely?

Yes, calculus can be used to predict the time it will take for a container to drain completely. By setting the volume function equal to zero, we can find the time at which the container will be empty. This is known as finding the root of the function and can be solved using various calculus techniques such as the Newton-Raphson method or the bisection method.

## 3. What factors affect the rate of water draining in a container?

The rate of water draining in a container can be affected by several factors, such as the size and shape of the container, the size of the drainage hole, the viscosity of the liquid, and the force of gravity. These factors can all be taken into consideration when using calculus to analyze the rate of water draining.

## 4. How can calculus be used to optimize the draining process?

Calculus can be used to optimize the draining process by finding the maximum rate of draining. This can be done by finding the maximum value of the derivative of the volume function, which represents the fastest rate of change in the volume of water. This maximum rate can then be used to determine the optimal size and placement of the drainage hole for the quickest draining time.

## 5. Can calculus be used to analyze the rate of water draining in real-life situations?

Yes, calculus can be used to analyze the rate of water draining in real-life situations. It is a powerful tool for understanding and predicting the behavior of fluids and can be applied to various scenarios, such as draining a swimming pool, filling and emptying a bathtub, or even modeling the flow of fluids in a river or pipe system.