The average ticket price for a concert at the opera house...

The average ticket price for a concert at the opera house was $550 .$ The average attendance was $4000 .$ When the ticket price was raised to $\$ 52,$ attendance declined to an average of 3800 persons per performance. What should the ticket price be to maximize the revenue for the opera house? (Assume a linear demand curve.)

Key Concepts

Revenue Function

The revenue function represents the total income generated from sales, calculated as the product of the unit price and the quantity sold. In problems involving pricing and attendance, this function is used to quantify how changes in price directly affect the overall revenue, which is essential for finding the price that maximizes income.

Linear Demand Curve

A linear demand curve is an assumption that the quantity sold (or attendance) changes at a constant rate with respect to changes in price. This simplification allows one to model the relationship between price and demand with a linear equation, making it easier to analyze and optimize the revenue function in a straightforward mathematical manner.

Quadratic Optimization

Quadratic optimization involves finding the maximum or minimum of a quadratic function, which is common when the revenue function, derived from a linear demand model, becomes quadratic. This concept is crucial as it provides the mathematical framework and techniques, such as completing the square or using calculus, to determine the optimal price that maximizes revenue.

Vertex of a Parabola

When a quadratic function represents revenue, its graph forms a parabola. The vertex of this parabola represents the maximum revenue point when the function opens downward. Identifying the vertex using the formula -b/(2a) or through differentiation is a key concept in determining the optimal pricing strategy to achieve maximum revenue.

Transcript

00:01 We're told in this problem that we have an average ticket price for a concert at the opera house, and it was initially $50.

00:10 And when we were charging $50 a ticket, we got our attendance was 4 ,000 people.

00:18 And then we raised the tickets prices to $52, and we got fewer people coming.

00:26 And this time we got 200 fewer people or 3 ,800 people coming to the...

00:32 To the upper house.

00:34 So what we want to do is figure out if we have a linear supply demand curve, where would we be to get maximum profit? so what happens is, i mean, if you just make a little plot here, you know, attendance and price, you got one point at 50, let's say, let's move that out here, and one another point at 52.

01:04 So at 52, you had a slightly lower attendance, and at 50 you had a slightly greater attendance.

01:14 So your attendance, you're assuming that you basically have a linear supply and demand curve, so that as you, as you, you know, decrease the price, more people were 10, which makes sense.

01:27 But the fact that it's linear is, you know, that's a big assumption.

01:32 Obviously, you know, this is going to probably, you know, in the small region here, it will certainly be linear.

01:42 But again, it's really hard to figure out.

01:47 But if you only have two data points, that's the best you can do is fit a line through those two things.

01:52 And so i did fit a line through those.

01:55 And it turns out that the attendance is 9 ,000 minus 100 times.

02:03 The price.

02:05 So obviously you can see that in the extreme when the tickets are free, you would be getting 9 ,000 people, which, you know, obviously if the tickets were free, well, i mean, maybe only 9 ,000 people would know about it, or maybe 9 ,000 people would care, even if the tickets were free.

02:23 But again, there's lots of other constraints in these problems, but we're just doing a problem here to see what we should, kind of which way we should be moving the price.

02:35 So what the thing is is what we want to do is maximize our profit.

02:39 And the profit is the price times the attendance.

02:45 And so what we can see here is that what we need to do is just multiply the attendance curve by the price and that will be our profit.

02:54 So this is our profit as a function of the price.

02:59 And we want to maximize that...

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