A child flies a kite at a height of 90 feet, the wind carrying the kite horizontally away from the child at a rate of 30 feet per second. How fast must the child let out the string when the kite is 150 feet away from the child? The child must let out the string at a rate of 24 feet per second when the kite is 150 feet away from the child. So let's look at a picture. We know the height is 90 and the height is going to be a constant. The kite is being carried away from the child so both B&A are going to change. So if we know that A is given as 150 feet, we can find our B by using the Pythagorean theorem. 150 ^2 is going to equal 90 ^2 + b ^2 or B is 120. Now the next thing is we're going to use Pythagorean theorem to relate how A&B are changing in terms of time. So a ^2 equal 90 ^2 + b ^2. We take the derivative in terms of time. We get 2A DADT. The derivative of that 90 ^2 is a constant, so that's just zero. And the derivative of b ^2 is 2BD BDT. The twos will cancel on each side. We know that B is 120, we're given that DBDT is 30, and we know that A is 150. So DADT is going to be changing at 24 feet per second.