Q323 · Practice questionTechnology-freeComplex unfamiliar8 marks
QUESTION 323 (8 marks)
Visitors arrive at a night market at rate people per hour for . Let be the modelled number who have arrived by time , with .
a)[3 marks]
Determine the time and value of the maximum arrival rate.
b)[3 marks]
Show that .
c)[2 marks]
The model predicts a limiting total of visitors. Determine the exact time by which half have arrived.
WORKED SOLUTION
8 marksPractice marking scheme
ANSWER
a) h, maximum people/h. b) As shown. c) h.
Worked solution
a) implies , so . The derivative changes from positive to negative and .
b) Integrate: . From , . Factor the expression to give .
c) gives (the nonnegative root for ). Solving gives the stated time.
Exact answers unless rounding is specified; equivalent forms accepted.
a) Differentiates and solves the stationary equation.
a) Establishes a maximum.
a) Finds the maximum rate.
b) Integrates the rate.
b) Uses the initial condition.
b) Factors to the required form.
c) Uses the correct square root.
c) Solves for the exact time.
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