Answer:
[tex]82\sqrt{21}\text{ or approximately 375.77 miles per hour}[/tex]
Step-by-step explanation:
Please refer to the diagram below. R is the radar station and x is the distance from the station to the plane.
We are given that the plane is flying horizontally at an altitude of two miles and at a speed of 410 mph. And we want to find the rate at which the distance from the plane to the station is increasing when it is five miles away from the station.
In other words, given da/dt = 410 and x = 5, find dx/dt.
From the Pythagorean Theorem:
[tex]a^2+4=x^2[/tex]
Implicitly differentiate both sides with respect to time t. Both a and x are functions of t. Hence:
[tex]\displaystyle 2a\frac{da}{dt}=2x\frac{dx}{dt}[/tex]
Simplify:
[tex]\displaystyle a\frac{da}{dt}=x\frac{dx}{dt}[/tex]
Find a when x = 5:
[tex]a=\sqrt{5^2-2^2}=\sqrt{21}[/tex]
Therefore, dx/dt when da/dt = 410, x = 5, and a = √(21) is:
[tex]\displaystyle \frac{dx}{dt}=\frac{(\sqrt{21})(410)}{5}=82\sqrt{21}\approx 375.77\text{ mph}[/tex]
The rate at which is distance from the plane to the radar station is increasing at a rate of approximately 375.77 miles per hour.
