Respuesta :
The rate at which the distance from the plane to the station is increasing when it is 5 mi away from the station is 421 mi/h.
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.
z = 1
[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 5 mi away from the station.
y = 5
According to Pythagoras
[tex]hypotenuse^2 = base^2 + perpendicular^2\\\\y^2 = x^2 + 1^2\\\\y^2 = x^2 + 1[/tex].... (1)
Put z=1 and y=5, to find the value of x.
[tex]5^2 = x^2 + 1^2\\\\25 = x^2 + 1\\\\25 - 1 = x^2\\\\24 = x^2[/tex]
Taking square root both sides.
[tex]\sqrt{24} = 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{24}[/tex], y=5, [tex]\frac{dx}{dt} = 430[/tex] in the above equation.
[tex]2\frac{dy}{dt} = \sqrt{24}(430)[/tex]
Divide both sides by 2.
[tex]\frac{dy}{dt} = \frac{\sqrt{24}(430) }{5} \\\\\frac{dy}{dt} = 421.312\\\\\frac{dy}{dt} = 421[/tex]
Hence the answer is the rate at which the distance from the plane to the station is increasing when it is 5 mi away from the station is 421 mi/h.
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