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Math Help - Related Rates cont'd...

  1. #1
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    Related Rates cont'd...

    I have some more questions that I need help setting up. Please feel free to switch out the numbers for variables so I can learn how to solve these. Thanks in advance for your help!

    Two towers (A and B) are 150m tall, 40m apart. A horizontal tightrope (height of 60m) is attached to the two buildings. A man walks across the tightrope from A to B at a constant 1/3 meter per sec. A spotlight is on the top of A. How fast is the shadow of the man moving up the wall of building B when he is 8 m away from B.

    (^^No clue where to begin^^)
    A trough, 12 feet long, has as its ends isosceles trapezoids with altitude 4 ft., lower base 3 ft., and upper base 4 ft. If water is let in at a rate of 7 cubic feet per minute, how fast is the water level rising when the water is 10 in. deep?

    (^^I found the volume of the trough to be 168 cubic feet ft...and I think the water level uses a ratio towards the trough...what next?^^)
    Stacey is brewing coffee that is being strained through a conical filter with a height of 12 in. diameter 8 in. The coffee flows from the filter into a cylindrical coffee pot with base area equal to 100pi square in. The depth, h, in inches, of the coffee in the conical filter is changing at the rate of (h-12) in. per min. How fast is the depth of the coffee in the cylindrical coffee pot changing when h=3in?

    (^^No clue where to begin^^)
    Two streets lights, each 30 feet tall, are 100 ft. apart. The light at the top of one of the poles is functioning properly, but the other light is burnt out. A repairman is climbing up the pole to fix the light at a rate of .5 foot per second. How fast is the tip of the repairman's shadow moving when he is 16 ft. up the pole?

    (^^I have a drawing set up, but I do not know how to label it and how to set up my equations^^)
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  2. #2
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    Hello, nivek516!

    Here's the first one . . .


    Two towers, A\text{ and }B. are 150m tall, 40m apart.
    A horizontal tightrope (height of 60m) is attached to the two buildings.
    A man walks across the tightrope at a constant \tfrac{1}{3} m/sec.
    A spotlight is on the top of A. How fast is the shadow of the man
    moving up the wall of building B when he is 8 m away from B]?
    First, make a sketch . . .
    Code:
        A *                       * B
          |  *                    |
       90 |     *                 |
          |        *      40-x    |
        T + - - - - - * - - - - - + R
          |     x     M  *        |
          |                 *     | y
       60 |                    *  |
          |                       * S
          |                       |
          |                       |
        C * - - - - - - - - - - - * D
                     40

    The buildings are AC\text{ and }BD\!:\;\;AC = BD = 150,\;CD = 40

    The tightrope is TR\!:\;\;TR = 40,\;TC = 60.\;AT = 90

    The spotlight is at A, the man is at M.
    . . Let: TM \:=\: x \quad\Rightarrow\quad MR \:=\:40-x
    . . We are given: . \tfrac{dx}{dt} = \tfrac{1}{3} m/sec.

    The man's shadow is at S\!:\;\;\text{let } RS = y.


    Since \Delta ATM \sim \Delta SRM\!:\;\;\frac{x}{90} \:=\:\frac{40-x}{y} \quad\Rightarrow\quad y \:=\:90\,\frac{40-x}{x} \:=\:90\left(40x^{-1} - 1\right)

    Differentiate with respect to time: . \frac{dy}{dt} \:=\:-\frac{3600}{x^2}\cdot\frac{dx}{dt}


    When MR = 8\;(x = 32)\!:\;\;\frac{dy}{dt} \;=\;-\frac{3600}{32^2}\left(\frac{1}{3}\right) \;=\;-\frac{75}{64}


    The shadow is moving up building B at 1\tfrac{11}{64}\text{ m/sec}

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  3. #3
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    Thanks! Could you explain why the answer is negative? Does it matter since it is speed? Any possible chance you could help me with the last two questions? I found out how to do the second one.
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  4. #4
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    Hello again, nivek516!

    Here's the last one . . .


    Two streets lights, each 30 feet tall, are 100 feet apart. The light at the top
    of one of the poles is functioning properly, but the other light is burnt out.
    A repairman is climbing up the pole to fix the light at a rate of 0.5 ft/sec.
    How fast is the tip of the repairman's shadow moving when he is 16 ft. up the pole?
    Code:
        A *               * C
          |   *           |
          |       *       |
          |           *   |
       30 |               * R
          |               |   *
          |              y|       *
          |               |           *
        B *---------------*---------------*
          : - - 100 - - - D - - - x - - - S

    The two poles are: . AB \,=\,CD\,=\,30
    . . and: . BD = 100

    The light at A shines on the repairman R and casts his shadow at S.
    . . Let: . x \,=\,DS,\;y \,=\,RD

    Since \Delta RDS \sim\Delta ABS\!:\;\;\frac{x}{y} \:=\:\frac{x+100}{30} \quad\Rightarrow\quad x \:=\:\frac{100y}{30-y}

    Differentiate with respect to time: . \frac{dx}{dt} \;=\;\frac{3000}{(30-y)^2}\,\frac{dy}{dt}

    Now substitute: . y = 16,\;\;\frac{dy}{dt} = 0.5 . . .

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