Page 50 - How To Solve Word Problems In Calculus
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4

                                                   ρ(x) = m (x) = 4x −1/2  = √
                                                                             x
                                                   ρ(4) = 2 slugs/ft

                                The movement of electrons through a wire produces an elec-
                                tric current. If Q is the charge flowing through the wire mea-
                                sured in coulombs, then the current I, measured in amperes,
                                is the rate of change of Q with respect to time in seconds.


                                EXAMPLE 10
                                The charge in coulombs that passes through a wire after t
                                seconds is given by the function

                                                                 2
                                                           3
                                                   Q(t) = t − 2t + 5t + 2
                                (a) Determine the average current during the first two
                                   seconds.
                                (b) Determine the current at the end of two seconds.

                                    Solution
                                         	Q     Q(2) − Q(0)    12 − 2
                                (a) I av =    =              =         = 5 amperes
                                          	t       2 − 0          2
                                                    2
                                (b) I(t) = Q (t) = 3t − 4t + 5

                                              2
                                   I(2) = 3(2) − 4(2) + 5 = 9 amperes
                                EXAMPLE 11
                                The number of bacteria in a Petri dish after t hours is
                                        3
                                              2
                                n(t) = 2t + 5t + t + 2. How fast is the population growing
                                after 3 hours?
                                    Solution

                                                     dn
                                                             2
                                                         = 6t + 10t + 1
                                                      dt
                                When t = 3, dn/dt = 85. The bacteria are growing at the rate
                                of 85 per hour.

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