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Related rates - finding hypotenuse of triangle

  1. Oct 29, 2016 #1
    1. The problem statement, all variables and given/known data
    A plane flying horizontally at an altitude of 3 mi and a speed of 480 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 4 mi away from the station. (Round your answer to the nearest whole number.)

    2. Relevant equations
    ##a^2=b^2+c^2## where ##a## is the hypotenuse of a triangle.

    3. The attempt at a solution
    I started by relating the given variables as follows
    • Altitude is a constant of 3 mi
    • Horizontal distance of the plane will be ##x##. We measure when ##x=4## miles.
    • Distance from the plane to the radar station will be ##y##. We measure when ##y^2=4^2+3^2 \longrightarrow y=5## miles.
    • Change in horizontal distance will be ##\frac{dx}{dt}## We are given that this is 480 miles/hour.
    We have ##y^2=x^2+3^2##. Taking the derivative with respect to time gives ##2y\times\frac{dy}{dt}=2x\times\frac{dx}{dt}+0##. Substituting in known values gives: ##2(5)\times\frac{dy}{dt}=2(4)(480) \longrightarrow \frac{dy}{dt}=\frac{4\times480}{5}=384##. Yet this is not the answer.

    Where am I going wrong? I've actually drawn out the triangle and variables, and I'm fairly stuck as to which part I'm messing up. Any insight will be appreciated!
     
  2. jcsd
  3. Oct 29, 2016 #2

    phinds

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    Gold Member
    2016 Award

    How did "4 mi away from the station" become only the horizontal component of how far the plane is away from the station?
     
  4. Oct 29, 2016 #3
    Looks like I totally misread the question. I thought that the problem statement stated 4 to be the distance traveled in the x direction, not the distance between the plane and the station. That's cleared everything up.

    Using the above information, the value for ##x## is ##\sqrt{4^2-3^2}=\sqrt{7}##. Plugging the correct values into my above derivative gives the correct answer to this problem: ##2(4)\times\frac{dy}{dt}=2(\sqrt{7})(480)\longrightarrow\frac{dy}{dt}=\frac{\sqrt{7}\times480}{4}\approx317##.

    Thanks for pointing that out!
     
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