The rate at which tickets were sold for [0, 9] is modeled by r(t)=550te-t/2 tickets per hour. Based on the model, how many tickets were sold by 9 P.M. (t=3) to the nearest whole number?

Question

Concert tickets went on sale at noon (t= 0) and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time t is modeled by a twice-differentiable function L for [0, 9]. …

The rate at which tickets were sold for [0, 9] is modeled by… r(t)=550te^-t/2 tickets per hour. Based on the model, how many tickets were sold by 9 P.M. (t=3) to the nearest whole number?

Use your calculus knowledge and show work.

Answer 1

The rate at which tickets are sold is in units of tickets per hour. Can you see that if you multiply the rate (tickets/hr) by the number of hours, you will get the number of tickets sold? But, it's not just a simple multiplication problem because the rate is constantly changing as time goes by. So, you must use Calculus to do the problem:

Since r(t) = 550te^-t/2 then the number of tickets sold is:

∫550te^-t/2 dt

and the number of tickets sold between t = 0 and t = 3 is:

₀∫³(550te^-t/2 dt

To do this integral manually, you would use Integration by Parts where u = t and dv = e^-t/2

Otherwise, you could use your calculator.