Question 1200637: The number of tickets sold each day for an upcoming performance of Handel's Messiah is given by N=−0.5x2+13x+12, where x is the number of days since the concert was first announced. When will daily ticket sales peak and how many tickets will be sold that day?
Answer by ikleyn(52848)   (Show Source):
You can put this solution on YOUR website! .
The number of tickets sold each day for an upcoming performance of Handel's Messiah
is given by N=−0.5x2+13x+12, where x is the number of days since the concert
was first announced. When will daily ticket sales peak and how many tickets
will be sold that day?
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The given function N(x) = −0.5x^2 + 13x + 12 is a parabola opened downward.
It has a maximum.
Its optimal value of x can be found using the formula  =  ,
where "a" is the coefficient at x^2 and "b" is the coefficient at x.
In our case, a= -0.5, b= 13, so =  =  = 13.
To find the maximum value of the function, substitute x= 13 into the formula N(x)
N(13) = -0.5*13^2 + 13*13 + 12 = 96.5.
At this point, turn on your attention: the number of tickets must be integer number,
but we got a non-integer value.
At this point, turn on your common sense and consider another integer argument in vicinity of 13,
which, possible, will give you integer number of N(x).
Try x= 12. You will have
N(12) = -0.5*12^2 + 13*12 + 12 = 96.
It is just much better and more reasonable.
So, the answer is: the daily ticket sales peak is 96 tickets on 12-th day.
Notice, that the other possible answer is 96 tickets on 14-th day, too.
Solved.
So, the solution is a standard algebraic procedure, but we should turn on our common sense
and make the necessary corrections in order for the solution be meaningful.
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On finding the maximum/minimum of a quadratic function see the lessons
- HOW TO complete the square to find the minimum/maximum of a quadratic function
- Briefly on finding the minimum/maximum of a quadratic function
- HOW TO complete the square to find the vertex of a parabola
- Briefly on finding the vertex of a parabola
Consider these lessons as your textbook, handbook, tutorials and (free of charge) home teacher.
Learn the subject from there once and for all.
Also, you have this free of charge online textbook in ALGEBRA-I in this site
- ALGEBRA-I - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".
Save the link to this online textbook together with its description
Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson
to your archive and use it when it is needed.
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