A charter flight charges a fare of 200 per person, plus 4...

A charter flight charges a fare of $\$ 200$ per person, plus $\$ 4$ per person for each unsold seat on the plane. If the plane holds 100 passengers and if $x$ represents the number of unsold seats, find the following. (a) A function defined by $R(x)$ that describes the total revenue received for the flight (Hint: Multiply the number of people flying, $100-x,$ by the price per ticket, $200+4 x$.) (b) The graph of the function from part (a) (c) The number of unsold seats that will produce the maximum revenue (d) The maximum revenue

A charter flight charges a fare of $\$ 200$ per person, plus $\$ 4$ per person for each unsold seat on the plane. If the plane holds 100 passengers and if $x$ represents the number of unsold seats, find the following.
(a) A function defined by $R(x)$ that describes the total revenue received for the flight (Hint: Multiply the number of people flying, $100-x,$ by the price per ticket, $200+4 x$.)
(b) The graph of the function from part (a)
(c) The number of unsold seats that will produce the maximum revenue
(d) The maximum revenue

Key Concepts

Graphical Representation of Quadratic Functions

A quadratic function can be plotted as a parabola. The direction in which the parabola opens (upward or downward) depends on the sign of the leading coefficient. This concept is crucial for visualizing the behavior of the function and identifying key features such as the vertex, which corresponds to the maximum or minimum value.

Optimization of Quadratic Functions

Optimization in the context of quadratic functions involves finding the maximum or minimum value of the function, which is typically achieved by locating its vertex. This is an important concept in various applications, such as determining the optimal number of units sold to maximize revenue or minimize cost.

Revenue Function Modeling

This concept involves setting up an expression that calculates total revenue by considering the number of units sold and the price per unit. It often requires identifying relationships between variables and translating conditions from a real-world context into a mathematical equation.

Quadratic Functions

When the revenue function is expanded, it forms a quadratic expression, which is a polynomial of degree two. Understanding quadratic functions is essential because they have properties such as symmetry, a vertex, and predictable maximum or minimum values based on the leading coefficient.

Transcript

00:02 I need to find a function that describes the revenue coming in from a charter flight, and then i want to maximize that revenue.

00:11 Before i write the function, though, let's make sure we understand all of the variables.

00:16 X is the number of unsold seats.

00:23 Since there are 100 possible seats on this plane, 100 minus x tells me the number of flying passengers.

00:37 Now, how much is the cost per ticket? that's the third piece that we need to make sure we know.

00:42 The cost per ticket is a flat $200 plus $4 per person for each unsold seat.

00:52 So i'm going to have to add 4x to that baseline of $200.

00:57 Okay.

00:58 How much revenue is coming in for this flight? the money coming in is the number of passengers times the cost for each of their tickets.

01:07 That's 100 minus x times 200 plus 4.

01:13 Let's get rid of those parentheses.

01:17 That gives me 20 ,000 plus 200x, minus 4x squared.

01:25 And i'm going to rewrite that so that all of my terms are in descending order.

01:31 So that's minus 4x squared plus 200x plus 20 ,000.

01:38 So that is the function that shows the revenue for my flight.

01:43 Now if you look at the x squared term, you can see that it has a coefficient.

01:46 Of negative 4.

01:48 A negative coefficient means this is a downward facing parabola.

01:53 So the vertex is the maximum value for my function.

01:58 So at the vertex, the x, the number of unsold seats is going to maximize my revenue.

02:06 So i want to find what that number is.

02:09 What x gives me my maximum value of y? to do that, we are going to complete the square.

02:18 Every term with an x, i'm going to keep together.

02:23 Every term without an x, in this case 20 ,000.

02:27 I'm going to push off to the side by itself for right now.

02:31 To complete the square, x squared should have a coefficient of 1.

02:35 So i'm going to factor out a minus 4 from every term that has an x in it.

02:42 And again, that 20 ,000 is just going to sit by itself for the moment.

02:46 Now to complete the square.

02:48 My x term has a coefficient of negative 50.

02:51 I need to take half of the coefficient and square it.

02:57 That's the number i'm going to put back up into my function.

03:00 So i'm going to add 625...

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