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Related Rates Sand Pile Problem

  1. Jan 11, 2016 #1
    1. The problem statement, all variables and given/known data
    A machine starts dumping sand at the rate of 20 m3/min, forming a pile in the shape of a cone. The height of the pile is always twice the length of the base diameter. After 5 minutes, how fast is the height increasing? After 5 minutes, how fast is the area of the base increasing?

    2. Relevant equations

    V = (1/3)(pi)r^2(h)

    3. The attempt at a solution

    For the first question,
    ##h = 4r##
    ##\displaystyle V(t) = 20t = \frac{1}{3}\pi r^2 h = \frac{\pi}{48}h^3##
    From this equation I get
    ##\displaystyle h = (\frac{960}{\pi}t)^{\frac{1}{3}}##
    then
    ##\frac{dh}{dt} = \frac{320}{\pi ((\frac{960}{\pi})t)^{\frac{2}{3}}}##
    When 5 is substituted for t, I get 0.77 m^3/min. Is this correct?

    Also, I am not sure how to approach the second problem. I know that the area of the base is ##\pi r^2##, but I am not sure how to proceed...
     
    Last edited: Jan 11, 2016
  2. jcsd
  3. Jan 11, 2016 #2
    Use the fact that height is equal to twice the diameter of the base at all times.
     
    Last edited: Jan 11, 2016
  4. Jan 11, 2016 #3
    I did that for the first problem. I don't see how it will work for the second problem though.
     
  5. Jan 11, 2016 #4
    EDIT: ah never mind, fairly straightforward problem. Haven't seen one like this in awhile though.

    Method 1: If you're know what dH/dt is, try applying that to see what the radius would be.

    EDIT: you did the first part right.
    Suggested Hint: Try rewriting the area equation.

    ...and check your units to part 1.
     
    Last edited: Jan 11, 2016
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