calculus 1 question

Question

a hot air balloon full of seniors skipping school rising from the ground of the parking lot at 140 ft/min. The teacher is standing at the door 500 feet from the point of lift off. When the balloon is 100 feet above the ground, find the rate at which the distance from the teacher to the balloon is changing.

Answer 1

Make a sketch:

Let "s" be the distance to the teacher and let "h" be the height of the balloon.

You can write a relationship between the parts of this by using the Pythagorean Theorem:

500² + h² = s²

You can solve for "s":

s = (250000 + h²)^1/2

Now you can take the derivative with respect to time of both sides (but you must use the chain rule):

ds/dt = 1/2(250000 + h²)^-1/2•(2h•dh/dt)

We are told that h = 100 and dh/dt = 140

ds/dt = 1/2(250000 + 100²)^-1/2•(2(100)•(140))

ds/dt = 1/2(260000)^-1/2•(28000)

ds/dt = 27.456 ft/min