Calculus

An airplane flys at a constant altitude of 2 miles and a constant speed of 600 miles per hour on a straight course that will take it directly over a kangaroo on the ground. How fast is the angle of elevation of the kangaroo's line of sight increasing when the distance from the kangaroo to the plane is 3 miles? Give your answer in radians per minute.

asked by Hailey
  1. I made a sketch showing the distance covered by the plane after it passed over the kangaroo as x miles
    let the angle of elevation be Ø

    I have :
    tanØ = 2/x
    xtanØ = 2
    x sec^2 Ø dØ/dt + tanØ dx/dt = 0

    when the distance between the kangaroo and the plance is 3
    x^2 + 2^2 = 3^2
    x = √5

    when x = √5 , dx/dt = 600, tanØ = 2/√5, and sec^2 Ø = 49/5
    √5(49/5) dØ/dt + (2/√5)(600) = 0
    dØ/dt = (-1200/√5)/(49√5/5)
    = -1200/49 radians/hr
    = -20/49 radians/min

    notice the angle is decreasing
    check my arithmetic, I should write it out on paper first.

    posted by Reiny
  2. tan(Θ) = 2/x
    dx/dt = -600
    y = 2

    Start with tan(Θ) = 2/x.Taking the derivative leaves you with:

    sec^2(Θ)*dΘ/dt = -2/x^2*dx/dt

    Solve for dΘ/dt.

    dΘ/dt = -2/x^2 * dx/dt * 1/sec^2(Θ)

    when the distance from the plane to the kangaroo is 3, x = sqrt(5); so Θ = arcsin(2/3).

    Use the equation above and plug in dx/dt, x, and theta, then divide by 60 to get units to rads/min.

    the answer should be about 2.22222 rad/min.

    posted by David Kelman

Respond to this Question

First Name

Your Answer

Similar Questions

  1. calculus

    An airplane flys at a constant altitude of 2 miles and a constant speed of 600 miles per hour on a straight course that will take it directly over a kangaroo on the ground. How fast is the angle of elevation of the kangaroo's line
  2. Math

    An airplane flys at a constant altitude of 2 miles and a constant speed of 600 miles per hour on a straight course that will take it directly over a kangaroo on the ground. How fast is the angle of elevation of the kangaroo's line
  3. Math

    An airplane flys at a constant altitude of 2 miles and a constant speed of 600 miles per hour on a straight course that will take it directly over a kangaroo on the ground. How fast is the angle of elevation of the kangaroo's line
  4. Math

    An airplane in Australia is flying at a constant altitude of 2 miles and a constant speed of 600 miles per hour on a straight course that will take it directly over a kangaroo on the ground. How fast is the angle of elevation of
  5. Calculus

    Derivatives - chain rule An airplane, flying horizontally at an altitude of 1 mile, passes directly over an observer. If the constant speed of the airplane is 400 miles per hour, how fast is its distance from the observer
  6. Calculus

    Derivatives - chain rule An airplane, flying horizontally at an altitude of 1 mile, passes directly over an observer. If the constant speed of the airplane is 400 miles per hour, how fast is its distance from the observer
  7. Calculus

    An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of the plane is 600 miles per hour. Find the rate at which the angle of elevation tetra is changing when the angle is 30 degrees
  8. Calculus

    An airplane flies at an altitude of y = 5 miles toward a point directly over an observer. The speed of the plane is 500 miles per hour. Find the rates at which the angle of elevation θ is changing when the angle is θ = 45°, θ
  9. calculus

    An airplane flies at an altitude of 5mies toward a point directly over an observer. The speed of the plain is 600 miles per hour. Find the rate at which the angle of elevation thetha is changing when the angle is 75 degrees.
  10. calculus

    an airplane is flying at an altitude of 6.7 miles towards a point directly over an observer. if the speed of the plane is 499 miles per hour, find the rate at which the angle of observation, Ɵ, changing by at the moment when

More Similar Questions