# Finding volume of swimming pool

## Homework Statement

A swimming pool is circular with a 40-ft. diameter. The depth is constant along east-west lines and increases linearly from 2 ft.at the south end to 7 ft. at the north end. Find the volume of water in the pool.

## Homework Equations

Dont know how to enter integrals here, but we need to use the polar coordinates here.

Refer to this: http://tutorial.math.lamar.edu/Classes/CalcIII/DIPolarCoords.aspx

## The Attempt at a Solution

I know how to integrate and find the volume. I am having trouble decomposing the problem into a picture and finding the limits.

SammyS
Staff Emeritus
Homework Helper
Gold Member
What's the average depth of the pool?

DaveC426913
Gold Member
Bah.

Dump a known quantity of dye or marker chemical in the pool. Run the pumps till it's evenly distributed. Take a water sample and test the concentration of the dye. Poof. You have the volume.

Works on closed bodies of water of any size & shape. hunt_mat
Homework Helper
I think that there is an easier way. Take the shape and flip another one and place it underneath it so the two shapes form a perfect cylinder. As yourself, what will be the height of the cylinder? Once you know this, you can easily calculate the volume of the cylinder which will be twice the volume of the original pool in question.

Redbelly98
Staff Emeritus
Homework Helper
We have two winning entries: What's the average depth of the pool?

I think that there is an easier way. Take the shape and flip another one and place it underneath it so the two shapes form a perfect cylinder. As yourself, what will be the height of the cylinder? Once you know this, you can easily calculate the volume of the cylinder which will be twice the volume of the original pool in question.

DaveC426913
Gold Member
Or simply duplicate the sloped section, then halve it.

So, one cylinder that's 2 ft high, plus one cylinder that's 7-2=5 ft high, which will be halved.

Question: if if this a calculus problem, are these solutions cheating?

SammyS
Staff Emeritus
Homework Helper
Gold Member
...
Question: if if this a calculus problem, are these solutions cheating?

Probably, but OP has not returned to the scene of the crime!

hunt_mat
Homework Helper
Okay, here is my idea, the has to be split up into two parts, that which is just a cylinder and that with the sloped part. I think that the key is to come up with the equation for the slope, and this will be $z=z(r)$, so if we sit our cylinder so the centre of it is is sat at $r=0$ then the equation of the slope will be:
$$\frac{5}{40}=\frac{z-0}{r-20}$$
So this gives us a $z=z(r)$, which we can invert to give $r=r(z)$, and the integral becomes:
$$V=\int_{0}^{5}\int_{0}^{2\pi}\int_{0}^{r(z)}rdrd \theta dz$$

Or something like that....

Redbelly98
Staff Emeritus