what is the volume of cylinder if The total surface area of a cylinder is $80\pi~\text{cm}^2$ and the difference between the 
height and the radius is $2~\text{cm}$.  What is the volume of that cylinder?
I have tried to find the height with the help of area but I can't.
 A: If the area includes the top and bottom disks we have
$$ 80 \pi = 2 \pi r^2 + 2 \pi rh \Rightarrow r(r+h) = 40 $$
if we have $h=r+2$ then $ 2 r^2 + 2 r = 40 $ 
then $r$ is the positive solution to $r^2+r-20=0$ ( which happens to be $r=4$cm )
so $h=r+2 = 6$cm.
So $V=\pi r^2 h = 96 \pi \; {cm}^3$
A: Since you did not provide any attempt at solving your problem, I will only give you a (strong) hint. Let $r$ be the radius and $h$ be the height. Then the total area is
$$
80\pi = 2\pi r^2 + 2\pi rh.
$$
You also know that $h - r = 2$. That is $h = r+2$. You want to find the volume $\pi r^2 h$ and you can do this by first findind $r$ and $h$. Using that $h = r+2$ in the first equation you get
$$
80\pi = 2\pi r^2 + 2\pi r (r+2).
$$
Left for you to do is to solve this equation for $r$.
A: $ h-r=2$ and $ 2 \pi r^2 + 2 \pi h= 80 \pi $
A: Let the height of cylinder be $h$ & radius $r$ then as given conditions we have $h-r=2 cm$ &
If the area is $2\pi r^2+2\pi rh=80\pi$ then we have $$2\pi r^2+2\pi r(r+2)=80 \pi => r^2+r-20=0$$ On solving above quadratic equation for $r>0$, we get $r=4 cm$ & thus height $h=6cm$ Hence the volume, 
$$=\pi r^2h=\pi(4^2)(6)=96 \pi \space cm^3$$ 
But if the given area is $2\pi rh=80\pi \space cm^2$ (As you have not mentioned which area is given) then we have $$rh=40 => h=\frac{40}{r}$$ Substitute this value in first relation, we get $$\frac{40}{r}-r=2$$ $$r^2+2r-40=0$$ Solving the quadratic equation for $r$, we get $$r=\frac{-2\pm \sqrt{(-2)^2-4(1)(-40)}}{2}=-1\pm \sqrt{41}$$ Since, $r>0$ hence, we get $r=\sqrt{41}-1$ & height $h=\sqrt{41}+1$ 
Hence, the volume of the cylinder $$=\pi r^2h=\pi (\sqrt{41}-1)^2(\sqrt{41}+1)=40\pi (\sqrt{41}-1)\approx 678.9766164 \space cm^3$$
