# Thread: Drone Aeroplane distance changing at an instant

1. ## Drone Aeroplane distance changing at an instant

A drone aeroplane is flying horizontally at a constant height of 4000 ft above a fixed radar tracking station. At a certain instant the angle of elevation theta is 30 degrees and is decreasing, and the speed of the drone aeroplane is 300 mi/h. How fast is the distance between the aeroplane and the radar station changing at this instant? Express the rate in units of ft/s. Use 1 mi=5280 ft

2. ## Re: Drone Aeroplane distance changing at an instant

Originally Posted by titansfreak93
A drone aeroplane is flying horizontally at a constant height of 4000 ft above a fixed radar tracking station. At a certain instant the angle of elevation theta is 30 degrees and is decreasing, and the speed of the drone aeroplane is 300 mi/h. How fast is the distance between the aeroplane and the radar station changing at this instant? Express the rate in units of ft/s. Use 1 mi=5280 ft
sketch a diagram?

let $\displaystyle z$ = straight-line distance between radar site and drone

let $\displaystyle x$ = horizontal distance between radar site and drone

let $\displaystyle h$ = plane's altitude (note that the altitude is in feet)

note when $\displaystyle \theta = 30^\circ$ , $\displaystyle z = 2h$ and $\displaystyle x = h\sqrt{3}$

$\displaystyle \frac{dx}{dt} = 300 \, mph$

$\displaystyle x^2 + h^2 = z^2$

take the time derivative of the above equation, substitute in your known (and calculated) values, and determine $\displaystyle \frac{dz}{dt}$

pay attention to units