Lola L.

asked • 07/12/17

Help me answer this Quadratic/ calculus problem Please

Starlight Production is hosting an annual outdoor concert. The average attendance is 40000 people when tickets are $100. For every $5 increase in ticket price, 1000 fewer people are expected to come to the concert.
 
a) Write a Quadratic function in standard form that represents the revenue earned. Be sure to indicate what the Variables represent.
 
b) What admission price will produce the most revenue? Explain how this can be determined.

Ann P.

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04/10/22

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Philip P. answered • 07/12/17

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Let x = the number of $5 increases in the ticket price:
 
     Ticket Price = $100 + $5x
     Attendance = 40,000 - 1000x
 
a)     Revenue = Ticket Price * Attendance
              = (100 + 5x)*(40,000 - 1000x)
              = 4,000,000 - 100,000x + 200,000x - 5000x2
        Revenue = -5000x2 + 100,000x + 4,000,000
 
b)  Take the derivative of Revenue wrt to x, set it to zero and solve for x.  The admission (ticket) price that maximizes Revenue will be $100 + $5x, where x is the value you just computed.  The answer is $150.
 
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Lola L.

What do you mean by the derivative of Revenue wrt to x?
 
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07/12/17

Philip P.

I assumed this was a calculus course (this is a standard type of business calculus problem).  If it's not calculus, you can find the answer algebraically.  The Revenue function is a quadratic equation.  Since the coefficient of the x2 term (-5000) is negative, the graph will be an inverted parabola with the vertex at the top.  The vertex is the point of maximum revenue.  For a quadratic of the general form y = ax2 + bx + c, the x coordinate of the vertex is x = -b/(2a).  In your equation, a = -5000 and b = 100,000.  Find the x value of the vertex, then plug that x value into the Ticket Price equation to find the Ticket Price that gives the max revenue.  The answers are x = 10 and Ticket price = $150.
 
h = (-100,000)/(-10,000)
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07/12/17

Philip P.

1) Take the derivative of Revenue wrt x:
 
     d(Revenue)/dx = d(-5000x2 + 100,000x + 4,000,000)/dx = -10,000x + 100,000
 
2) Set it to zero, solve for x:
     0 = -10,000x + 100,000
     10,000x = 100,000
     x = 10
 
3) Ticket Price = $100 + 5x = $100 + $50 = $150
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07/12/17

Lola L.

Thank You!
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07/13/17

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