Answer:
The rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station is 372 mi/h.
Step-by-step explanation:
Given information:
A plane flying horizontally at an altitude of "1" mi and a speed of "430" mi/h passes directly over a radar station.
[tex]z=1[/tex]
[tex]\frac{dx}{dt}=430[/tex]
We need to find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.
[tex]y=2[/tex]
According to Pythagoras
[tex]hypotenuse^2=base^2+perpendicular^2[/tex]
[tex]y^2=x^2+1^2[/tex]
[tex]y^2=x^2+1[/tex] .... (1)
Put z=1 and y=2, to find the value of x.
[tex]2^2=x^2+1^2[/tex]
[tex]4=x^2+1[/tex]
[tex]4-1=x^2[/tex]
[tex]3=x^2[/tex]
Taking square root both sides.
[tex]\sqrt{3}=x[/tex]
Differentiate equation (1) with respect to t.
[tex]2y\frac{dy}{dt}=2x\frac{dx}{dt}+0[/tex]
Divide both sides by 2.
[tex]y\frac{dy}{dt}=x\frac{dx}{dt}[/tex]
Put [tex]x=\sqrt{3}[/tex], y=2, [tex]\frac{dx}{dt}=430[/tex] in the above equation.
[tex]2\frac{dy}{dt}=\sqrt{3}(430)[/tex]
Divide both sides by 2.
[tex]\frac{dy}{dt}=\frac{\sqrt{3}(430)}{2}[/tex]
[tex]\frac{dy}{dt}=372.390923627[/tex]
[tex]\frac{dy}{dt}\approx 372[/tex]
Therefore the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station is 372 mi/h.
