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Math Help - Confusing Rates of Change

  1. #1
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    Exclamation Confusing Rates of Change

    Hi

    How do I do this:

    A lamp is 40 m above the ground and a stone is dropped from the same height at a point 10 m away. Find the speed of the shadow on the ground given that in t seconds the stone falls 4.9t^2 m.

    Thanks a lot!
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  2. #2
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    Hello Sunyata
    Quote Originally Posted by Sunyata View Post
    Hi

    How do I do this:

    A lamp is 40 m above the ground and a stone is dropped from the same height at a point 10 m away. Find the speed of the shadow on the ground given that in t seconds the stone falls 4.9t^2 m.

    Thanks a lot!
    Study the diagram I have attached. The distance x m represents the distance of the shadow from C, the point directly below A, where the stone was dropped.

    Using similar triangles:
    \frac{RC}{BC}=\frac{RQ}{PQ}
    What you need to do now:
    Substitute into this equation the values I've given you in the diagram.

    Solve for x in terms of t.


    Differentiate to find \frac{dx}{dt}. This represents the speed of the shadow.

    Can you complete this now?

    Grandad
    Attached Thumbnails Attached Thumbnails Confusing Rates of Change-untitled.jpg  
    Last edited by Grandad; April 9th 2010 at 07:51 AM. Reason: Correction to diagram
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  3. #3
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    Exclamation

    Ummm am I meant to get 4.9t^2x + 49t^2 = 400???

    if so, when i implicitly differentiate that, I can't seem to get the answer...
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  4. #4
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    Hello Sunyata
    Quote Originally Posted by Sunyata View Post
    Ummm am I meant to get 4.9t^2x + 49t^2 = 400???

    if so, when i implicitly differentiate that, I can't seem to get the answer...
    See step 2 above:
    Solve for x in terms of t
    which gives
    x = \frac{400-49t^2}{4.9t^2}
    =\frac{400}{4.9t^2}-10
    \Rightarrow \frac{dx}{dt} = \frac{-800t^{-3}}{4.9}
    So the speed of the shadow has magnitude \frac{8000}{49t^3}.

    Grandad
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