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Derivative as a rate of change exercise

  1. Oct 15, 2014 #1
    1. The problem statement, all variables and given/known data

    A police car is parked 50 feet away from a wall. The police car siren spins at 30 revolutions per minute. What is the velocity the light moves through the wall when the beam forms angles of: a) α= 30°, b) α=60°, and c) α=70°?

    This is the diagram:
    • α is the angle the light beam makes with the siren (and it is a variable)
    • dα/dt is the angular velocity of the light beam (and it is a constant; 30 revolutions per minute)
    • x is the length of what the beam hits in the wall (it is a variable)
    • dx/dt is velocity on how much light is being struck in the wall (it is a variable, and what's we are supposed to compute with the given data)
    • Also we have the distance between the police car and the wall, which is also a constant and is 50 feet.
    Ok, before going to the formulas, according to my textbook the answers should be:
    • When the angle is 30° (= 0.524 rad) the velocity is 200π/3 feet per second
    • When the angle is 60° (= 1.047 rad) the velocity is 200π feet per second
    • When the angle is 70° (= 1.222 rad) the velocity is approximately 427.43π feet per second

    2. Relevant equations

    Well, this is a calculus problem, it does not take a pre-shaped formula as in physics or chemistry. However, we are dealing with derivatives as a rate of change. I found the relationship which may lead to the problem answer:

    If we consider the previous diagram the formulas are:
    • x=50tan(α) which leads to the one below
    • dx/dt=50(sec2α)(dα/dt)= 50(1/cos2α) (this is the equation I used for attempting to solve the problem)
    NOTE: For getting the second equation, I used the trigonometric derivatives rules

    3. The attempt at a solution

    I did use the last formula; replacing with the given data we get:

    For the angle of ,524 rad
    dx/dt=50(1/cos2(,524))=66.69 ft/s which is not the same as 200π/3 ft/s
    For the angle of 1,047 rad dx/dt=50(1/cos2(,524))=201.96 ft/s which is not the same as 200π ft/s
    For the angle of 1,222 rad dx/dt=50(1/cos2(,524))=423.4 ft/s which is not the same as 427.43 ft/s

    I need to see what I'm doing wrong; I've been stuck in this problem for about an hour and a half :(

    Attached Files:

  2. jcsd
  3. Oct 15, 2014 #2


    Staff: Mentor

    In your formula for dx/dt, it looks like you have lost dα/dt at the end.
  4. Oct 15, 2014 #3


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    In addition to Mark44's comments, don't change everything to decimals. Use ##30^\circ = \frac \pi 6##radians and ##30##rpm##=\pi##rad/sec.
  5. Oct 15, 2014 #4
    Thanks everyone, I checked and I got it right.
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