Now this one you will have to work backwards with me. First lets have a look at the formula for the surface area of a cone. This is Pi R^2 + Pi R L. Now we know the surface area so we can set up this equation. I will use 3.14in place of pi. 3.14 x 144 + 3.14 x 12 x L = 1050 in squared. Now we just solve for L. So 3.14 x 144 +3.14*12 = 489.84 Which means that L= 1050-489.84 = 560.16. L=560.16. (I rounded the 490.074). Now L is the length of the slanted edge like in this image. So the height of the slant rounded to the nearest 10th of an inch is 560.2 in.
I hope this is correct and that it helped you.
-G