# Satellite Dish - Max Amount and Tilt

#### Jason

Staff member
Moderator
What is the max amount of water the dish will hold, expressed as a triple integral, for parabolic dish that is $$\displaystyle 3\, m$$ wide and $$\displaystyle \dfrac{1}{2}\, m$$ deep? Consider the fact that it's axis of symmetry has a tilt $$\displaystyle 25$$ degrees from the vertical.

Second, what would be the smallest tilt of the dish where it can contain no water?

#### MarkFL

##### La Villa Strangiato
Staff member
Moderator
Math Helper
I think the first thing I would do is find the equation of the paraboloid. A circular paraboloid with its vertex at the origin has the form:

$$\displaystyle z=a\left(x^2+y^2\right)$$

Now, when $$\displaystyle z=\frac{1}{2}$$ we require $$\displaystyle x^2+y^2=\left(\frac{3}{2}\right)^2$$ and this implies:

$$\displaystyle a=\frac{2}{9}$$

And so our paraboloid is:

$$\displaystyle z=\frac{2}{9}\left(x^2+y^2\right)$$

The surface of the water will be a plane...how can we find this plane?

anemone