Hey,
So the surface area of a cylinder is:
$\displaystyle SA_{cyl} = 2\pi r^2 + 2\pi rh$
Since we are taking off the top of the cylinder we can change it to:
$\displaystyle SA_{cyl} = \pi r^2 + 2\pi rh$
Next we need the surface area for a cone, minus the area of the base of the cone.
$\displaystyle SA_{cone} = \pi rs + 2\pi r^2$
Again, we don't need the base of the cone, so we get:
$\displaystyle SA_{cone} = \pi rs$
This formula requires a value 's' which is the diagonal length (straight dotted line in your picture). We get this from Pythagoras.
$\displaystyle s^2 = h^2 + r^2 \Rightarrow s = \sqrt{12^2 + 5^2} = 13$
There we are! We have everything we need. So, the surface area of our modified cylinder is:
$\displaystyle SA_{mod} = SA_{cyl} + SA_{cone} = (\pi r^2 + 2\pi rh) + (\pi rs)$
$\displaystyle SA_{mod} = (\pi 5^2 + 2\pi(5)(12)) + (\pi (5)(13)) = 660 cm^2$
At least, that's what I get. Hope it helps!
--
Dave