The y value is 120 (fixed).
The x value is x0+10t. I assume the kite is moving away from you.
So the angle theta is arctan(120/(x0+10t)).
A kite 120 feet above the ground is moving horizontally at a rate of 10 feet per second. At what rate is the angle between the string and the horizontal changing at the instant 240 feet of the string are out?
I know to draw a right triangle, but I don't know to solve it in terms of angle fata. Any help is very much appreciated. Could someone lead me in the right direction?
You've gotten the right triangle correct. At the instant the string is 240 ft long, the angle is 30 degrees by your argument. However you're trying to find the rate at which the angle is changing. Does the 30 degrees become 29 degrees in the next second or the next 2 seconds? That's where you'll have to take a derivative.