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Related Rates Application.

  1. Oct 8, 2011 #1
    1. The problem statement, all variables and given/known data

    A hemispherical dome has a diameter of 100m. A search light was placed at point A as shown at the middle of the dome at B. A balloon was released vertically at a velocity of 4m/s. How fast will the shadow of the balloon move alone the roof if it traveled 25 m vertically.

    Image link:http://bit.ly/rem8PG [Broken]


    2. Relevant equations



    3. The attempt at a solution

    I don't know how to start this. I dont understand the nature of the shadow and why did it bounce off the wall like as shown in the image.

    What relationship should i use? Please help me to solve this step by step. I'm willing to learn.
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Oct 8, 2011 #2

    Mark44

    Staff: Mentor

    The shadow isn't bouncing off the inside of the dome. That line is just a radius of the dome, from the light out to the hemispherical wall.
     
    Last edited by a moderator: May 5, 2017
  4. Oct 8, 2011 #3

    Oh, so using that the value of the shadow will have height of 25 m which is 50m from point B after the instant of travelling upwards. But what is the use of the velocity of the balloon? What variables can I use to get the speed of the shadow?
     
  5. Oct 8, 2011 #4

    Mark44

    Staff: Mentor

    The shadow travels along the curved surface of the dome. As the balloon rised 25 m. the shadow will move along an arc whose length is r * [itex]\theta][/itex].
     
  6. Oct 9, 2011 #5
    I can't get θ without knowing how long did the balloon fly over B. If I only how long it took then I would have a fixed value after it traveled solving an angle at point A and getting θ by sine-law and supplementary angles.
     
  7. Oct 9, 2011 #6
    This is what I have so far:

    Since we are looking for velocity of shadow
    let that be = d(h)/d(t)

    and height of shadow = h

    So Sinθ = h/50m
    derivative of this equation is:
    Cosθ(dθ/dt) = [50m(1)[d(h)/d(t)] - h(0)] / 2500



    I have no values for θ and (dθ/dt) if I can get this I can get the d(h)/d(t) which is the main requirement of this equation. Still I haven't used the 4m/s velocity of the balloon from point B, I know that has a significant role in solving this but I can't seem to use it. I tried to get it's height after the instant but there is no time given. Help guys?
     
  8. Oct 9, 2011 #7
    Thank you this helped me solve the equation. The speed of the shadow is 6.4m/s
     
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