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Related rates of a ladder sliding

  1. Nov 15, 2016 #1
    1. The problem statement, all variables and given/known data
    A 5 m ladder is sliding down the wall, and h is the height of the ladder's top at time t, and x is the distance from the wall to the ladder's bottom at time t.
    Given that h(0) = 4 at t = 0 seconds and dh/dt = 1.2m/s, and the ladder is 5m long find x(2) and dx/dt at t=2 seconds

    2. Relevant equations
    x^2 + h^2 = 5^2
    25 - 16 = x^2
    x = 3

    3. The attempt at a solution
    $$2x*\frac{dx}{dt} + 2h*\frac{dh}{dt} = 0$$
    since we know that dh/dt = 1.2 and x = 3:
    $$2(3)*\frac{dx}{dt} + 2(4)*(1.2)= 0$$
    $$\frac{-9.6}{6} = \frac{dx}{dt}$$
    $$\frac{dx}{dt} = -1.6$$

    I believe dx/dt is -1.6 at t = 0 seconds, but how do I use this information to find x(2) and dx/dt at 2 seconds? Am I wrong in approaching this problem?
     
  2. jcsd
  3. Nov 15, 2016 #2

    Mark44

    Staff: Mentor

    In your problem statement, dh/dt appears to be constant at 1.2 m/s (really, this is -1.2 m/s). You're given that h(0) = 4, so what will h be at t = 2 sec.? From that you can find x(2).
     
  4. Nov 15, 2016 #3
    h(2) = 1.6m and x(2) = 4.73?

    Is my initial approach wrong though? And why is it that I got $$\frac{dx}{dt} = -1.6$$ which is the same coefficient of h(2)? I get that from a logical perspective this problem is very easy but I still want to do the math behind it.
     
  5. Nov 15, 2016 #4

    Mark44

    Staff: Mentor

    I don't get these values. dh/dt = -1.2 m/sec, and h(0) = 4. How did you get h(2) = 1.6 m?
     
    Last edited: Nov 16, 2016
  6. Nov 16, 2016 #5
    Realized my mistakes were because I wasnt' careful enough
    Okay so h(2) = 1.6m as 4-2(1.2) = 1.6
    then we can use that to find x(2):
    √(25-1.6^2) = 4.7377

    And then i input these values in to:

    $$2(4.73)*\frac{dx}{dt}=2(1.6)(1.2)$$
    $$\frac{dx}{dt} = \frac{3.84}{2(4.737)}$$

    and i get 0.405 for dx/dt.

    Edit: changed this post as the previous was wrong
     
    Last edited: Nov 16, 2016
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