# Distance of the point $(a,b,c)$ to the plane $z=0$

I'm trying to solve a calculus problem, I need to find the mass of a cylinder, I'm close to the answer, I got $$8\pi$$ but it should be $$16\pi$$.

I think my mistake lies in the density function since it uses some linear algebra (?) ideas.

"The density at the point $$(a,b,c)$$ equals double of the distance of the point to the plane $$z=0$$"

At first I did $$\sqrt{a^2+b^2}$$ but that's the distance of a point to the $$z$$-axis not $$z$$-plane right?

If we want to find the distance of a point $$(a,b,c)$$ to the $$z$$-plane can we just lower this point until it reaches the $$z$$-plane? So we have the distance between $$(a,b,c)$$ and $$(a,b,0)$$? So we get $$\sqrt{z^2}=z$$

• The distance is $|c|$. – Bernard Apr 16 at 21:30
• $\sqrt{c^2}=|c|$ not $c$. – Trebor Apr 16 at 23:51
• Imagine the $x$-$y$ plane as the ground. How far off or under the ground is $(x,y,z)$? – amd Apr 17 at 0:33
• Thank you, so my density function was kinda correct (missed the absolute value), in the end my mistake calculating the mass was not in the density function but when converting my integrals to polar coordinates, I'm starting to work with solids now and the process is a little different – Electrolite Apr 17 at 1:10

## 1 Answer

As people corrected in the comments the distance of the point (a,b,c) to the z-plane is the distance of this point (a,b,c) to the point (a,b,0) which is: $$|c|$$.

I managed to calculate the mass of the solid correctly and my question is solved.