How fast is the balloon rising up?

Question

A hot air balloon is rising 200 ft above ground while a person viewing the balloon has a height of 6 feet has a view of 45 degrees to the hot air balloon. If the distance between the balloon and the dude is changing at a rate of 25 feet per minute. How fast is the balloon rising up?

Answer 1

Let s be the distance between the person and the balloon which is the hypotenuse of a 45° right triangle. Since the person is 6 feet tall, the height of the balloon is 194 feet above the line parallel to the ground. Since the angle is 45°, the horizontal distance between the person and the point directly below the balloon is also 194 feet. Let y = the vertical distance above the 6 foot.

Need to find dy/dt. ds/dt = 25 ft/min as given.

This forms a right triangle, with both the base and height equal to 194 feet.

sin(45°) = √2/2 = 194/s --> s= 194√2

s²= 194² + y²

2sds/dt = 2ydy/dt

dy/dt = (s/y)ds/dt = (194√2)/194 (25) = 25√2 ft/min