Hot air ballon 280 feet away as it rises up into the sky. at first, the balloon is at an angle of elevation of 12 degrees. a few minutes later, the balloon rises to angle of elevation of 60 degrees. what is the change in altitudes between steve's two observations?

Step-by-step explanation:

This is right triangle trig.  The reference angle is 12 in one case and 60 in the other, but the horizontal distance doesn't change in either one, and neither does what you are looking for, which is the height of the balloon in both cases of the angle differences.  And if you're looking for the difference in the height, you'll find both and subtract the smaller from the larger.  

The height is across from the reference angle and the horizontal distance is adjacent to the reference angle, so the trig identity you want is tangent.  Set up according to the angle measure of 12 degrees:

and

280 tan(12) = x

x = 59.5 ft

Now for the angle measuring 60 degrees:

and

280 tan(60) = x

x = 484.9

The difference between the two heights is 425.5 feet.