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Math Help - rates problem

  1. #1
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    rates problem

    Can anyone help with this rates problem?

    A light is on the ground, 15 m from a wall. A woman, 1.6 m tall, walks away from the light and towards the wall at 0.5 m/s. Calculate the rate of change of the length of her shadow on the wall when the woman is 10 m from the wall.

    I guess you would have to find the rate at which the tip of her shadow is moving? (I think)
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  2. #2
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    Hello, math619!

    A light is on the ground, 15 m from a wall.
    A woman, 1.6 m tall, walks away from the light and towards the wall at 0.5 m/s.
    Calculate the rate of change of the length of her shadow on the wall
    when the woman is 10 m from the wall.

    I guess you would have to find the rate at which the tip of her shadow is moving.
    . . Right!
    Code:
                                  *
                               *  |
                            *     |
                         *        |
                      *           | s
                   *  |           |
                *     |1.6        |
             *   x    |           |
          *-----------+-----------*
          : - - - -  15 - - - - - :

    From similar right triangles, we have: . \frac{s}{15} \:=\:\frac{1.6}{x}
    . . and we have: . s \:=\:24x^{-1}

    Differentiate with respect to time: . \frac{ds}{dt}\:=\:-\frac{24}{x^2}\ \frac{dx}{dt}

    We are given: . \frac{dx}{dt} = 0.5 . . . and "10 m from the wall" means x = 5

    . . Therefore: . \frac{ds}{dt} \:=\:-\frac{24}{5^2}(0.5) \;=\;\boxed{-0.48\text{ m/sec}}

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