A crop duster is flying towards a ground fire at a constant height of 200 meters above the ground

Question

A crop duster is flying towards a ground fire at a constant height of 200 meters above the ground. If the distance between the crop duster and the fire is decreasing at a rate of 40 meters per second when the crop duster is 500 meters from the fire, how fast is the crop duster flying?

Answer 1

At any point in time we have a right angle triangle:

Its vertical height is h = 200m,

its hypotenuse y is the direct distance between the duster and the fire,

and the remaining horizontal side x is the horizontal distance

between the duster and the fire.

While h is constant, both x and y are functions of time.

From Pythagorean theorem

y² = x² + h²

Differentiate w.r.t. time

2yy' = 2xx'

We are to calculate the speed x' at the point in time when

y = 500m

y' = 40m/s

x' = yy'/x = yy'/sqrt(y² - h²)

Plugging in given numbers

x' = 500 . 40 / sqrt(500² - 200²) = 43.7m/s