Here, the problem says that "attendance is linearly related to ticket price." That tells you that you will need an equation for a line at some point. Let's start there. The most common form you may have seen a linear equation in is this:
y = mx + b.
First, we need to set what our variables are. Whenever doing word problems, I always write down labels like these in a corner of my paper for future reference. Let's say:
x = ticket price (a number of dollars)
y = attendance (a number of spectators).
To finish the setup, let's glance back over the problem to make sure that we have a variable for each of the quantities we're supposed to work with. The core question is: "What ticket price would maximize revenue?" Well, we already have a variable for ticket price, but not one for revenue. [We know we need a variable for revenue because we're asked to maximize it--so it could vary!] Then let's add one more to our list:
R = revenue (a number of dollars).
Now, how does revenue relate to our other variables? Revenue is how much money you earn in total. In this problem, the stadium earns the ticket price for each spectator who attends--these correspond to the x and y we set up earlier! Then we get the relationship
Revenue = ticket price * number of spectators
-------> R = xy.
We've been asked for the ticket price that maximizes revenue, so we want to end up with an equation that only has the corresponding variables, that is, x and R. We can do this by substituting out the y in the equation above, using the linear equation y = mx + b from before.
y = mx + b
To find m and b for any line, we need two points. So, we're looking for two pairs of values of x (ticket price) and y (attendance). The problem says, "With the ticket price at $12 the average attendance has been 29000. When the price dropped to $9, the average attendance rose to 32000." This gives us the two points
Now, let's find m using the slope formula:
m = (y1 - y2) / (x1 - x2)
= (32000 - 29000)/(9 - 12)
Plug in m and one of the points (doesn't matter which) to find b:
y = mx + b
32000 = -1000(9) + b
32000 = -9000 + b
32000 + 9000 = b
41000 = b
So, the final form of the equation for the line is
y = -1000x + 41000.
Now, let's get back to revenue! At the end of the setup section, we said we wanted to substitute out y in the revenue equation, so let's do it:
R = xy
R = x(-1000x + 41000)
R = -1000x2 + 41000x
We can see we've ended up with an equation for a parabola. The minus sign on the x2 term means that the parabola points downward, so the maximum is located at the vertex. The vertex formula is
x = -b/(2a)
[where a and b are the coefficients from the form ax2 + bx + c]
So, the value of x that maximizes R is 20.5. Translated back into the terms of the problem, this means that the ticket price that maximizes revenue is $20.5. This is what we were asked to find!
Applied Problem Check:
We should be done, but we do need to check in a word problem that there isn't some real-world constraint that could make the mathematical answer incorrect, like if it says we should have a negative number of people. In this problem, we were told that the stadium only holds 64000 spectators, so we should check that the corresponding number of spectators for the chosen ticket price falls under that. Since we have x = $20.5 and want to find y = number of spectators, we can use the linear equation again:
y = -1000x + 41000
= -1000(20.5) + 41000
= -20500 + 41000
= 20500 spectators
This is a positive number of people that falls within the stadium's capacity of 64000, so we're good!