1.   d = 0.0034t2 − 0.52518t + 20.


     To find " t " when the depth "d" is equal to 1 cm, you need to solve the quadratic equation

         0.0034t2 − 0.52518t + 20 = 1.

     To solve this equation, first write it in the standard form and then use the quadratic formula.


     If you don't know how to use the quadratic formula, learn it from the lessons

          - Introduction into Quadratic Equations

          - PROOF of quadratic formula by completing the square

     in this site.



2.  Let " t " be the time for the larger pipe to fill the tank working alone.

    Then the time to fill the tank for the smaller pipe is (t+9) minutes.


    Now, in one minute the larger pipe fills        of the tank volume;

                       the smaller pipe fills   of the tank volume.


    Working together, the two pipes fill   +   of the tank volume per minute.


     From the other side,  the two pipes, working together, fill    of the tank volume, according to the condition.


    It gives you an equation

         +  = .


    To solve it, multiply both sides by 6t*(t+9). You will get

        6(t+9) + 6t = t*(t+9).


     Transform this equation to the standard form


         6t + 54 + 6t = t^2 + 9t

         t^2 - 3t - 54 = 0


     Now you can solve it by factoring

         (t-9)*(t+6) = 0.


     It has two roots t= -6  and  t= 9.

     Of these two roots, only positive t= 9 is the valid solution to the problem.


     ANSWER.  Large pipe can fill the tank in 9 minutes, working alone.

              Small pipe can fill the tank in (9+9) = 18 minutes.


     CHECK.   +  =  =  = .    ! Correct !

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