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Math Help - Water draining from two tanks.

  1. #1
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    Water draining from two tanks.

    Hi all,
    There is no indication of base so am wondering how to solve this sum. Thank you for taking the time.
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  2. #2
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    Lexington, MA (USA)
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    Re: Is this sum solvable?

    Hello, fleurdel!

    18. There are two tanks filled with water.
    They are being emptied by two different taps, each with a constant flow of water.
    Tank A is 5 cm taller than Tank B.

    Code:
          * - - - *
          |: : : :|
          | : : : |     * - - - - *
          |: : : :|     |: : : : :|
        x | : : : |     | : : : : |
          |: : : :|     |: : : : :| x-5
          | : : : |     | : : : : |
          |: : : :|     |: : : : :|
          *-------*     *---------*
              A              B

    The tap of Tank A was opened at 7:00 a.m.
    Tank A was empty at 1:00 p.m.

    The tap of Tank B was opened at 8:30 a.m.
    Tank B was empty at 12:30 p.m.

    The water level of both tanks were equal at 11 a.m.

    What is the height of tank A?

    \text{Let }a\text{ = rate of flow of the tap on Tank A, relative to the height of the water,}
    . . \text{measured in cm/hour.}

    \text{Let }b\text{ = rate of flow of the tap on Tank B, relative to the height of the watr,}
    . . \text{measured in cm/hour.}


    \text{In 6 hours, the height of Tank A went from }x\text{ cm to 0 cm.}
    . . \text{We have: }\:6a \,=\,x .[1]

    \text{In 4 hours, the height of Tank B went from }x-5\text{ cm to 0 cm.}
    . . \text{We have: }\:4b \,=\,x-5 .[2]

    Substitute [1] into [2]: . 4b \:=\:6a - 5 \quad\Rightarrow\quad 6a - 4b \:=\:5 .[3]


    At ll:00 a.m. the two tanks had the same height.

    Tank A: in 4 hours, the level dropped 4a\text{ cm.}
    . . The height was: . x - 4a\text{ cm.}

    Tank B: in 2.5 hours, the level dropped \tfrac{5}{2}b\text{ cm.}
    . . The height was: . (x-5)-\tfrac{5}{2}b\text{ cm.}

    We have: . x - 4a \:=\:(x-5) - \tfrac{5}{2}b \quad\Rightarrow\quad 8a - 5b \:=\:10 .[4]

    Solve [3] and [4]: . \begin{Bmatrix}6a - 4b &=& 5 \\ 8a - 5b &=& 10 \end{Bmatrix} \quad\Rightarrow\quad \begin{Bmatrix}a &=& \frac{15}{2} \\ b &=& 10 \end{Bmatrix}


    Substitute into [1]: . x \:=\:6a \:=\:6\left(\tfrac{15}{2}\right) \:=\:45

    Therefore, Tank A is 45 cm high.

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  3. #3
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    Re: Water draining from two tanks.

    Thank you so much, Soroban.
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