The distance between the Planes is changing at the rate of
6400 / [tex]\sqrt{89}[/tex] mph
The length of an object's route as a whole is its distance. The smallest distance between a position's starting point and ending point is called displacement. Displacement is a vector, whereas distance is a scalar. A distance travelled by an item can never be zero, although its displacement can.
x = Distance of plane 1 from airport
y = Distance of plane 2 from airport
So using Pythagorean theorm,
X² + Y² = L²
L is the distance between P1 and P2
[tex]\frac{d}{dt}[/tex] ( X² + Y² ) = [tex]\frac{d}{dt}[/tex] ( L² )
= 2x[tex]\frac{dx}{dt}[/tex] + 2y [tex]\frac{dy}{dt}[/tex] = 2L [tex]\frac{dL}{dt}[/tex]
= x[tex]\frac{dx}{dt}[/tex] + y [tex]\frac{dy}{dt}[/tex] = L [tex]\frac{dL}{dt}[/tex]
Given, x = 8 mi and y = 5 mi
[tex]\frac{dx}{dt}[/tex] = 400mph
[tex]\frac{dy}{dt}[/tex] = 640 mph
so, L² = [tex]\sqrt{8^2+ 5^2 \\[/tex] = [tex]\sqrt{89}[/tex]
So,
[tex]\frac{dL}{dt}[/tex] = x[tex]\frac{dx}{dt}[/tex] + y [tex]\frac{dy}{dt}[/tex] / L = 8 (400) + 5 (640) / [tex]\sqrt{89}[/tex]
[tex]\frac{dL}{dt}[/tex] = 6400 / [tex]\sqrt{89}[/tex] mph
To learn more about Pythagoras Theorm , visit:
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