I'm trying to understand the solution to this simple mathematical modelling question

Question

This is an example from Gary Chartrand's "Introductory Graph Theory" (Page 8, Example 1.5). He provides an answer without actually showing the steps or working, and I am unable to figure it out. The statement of the problem is (after editing out extraneous content):

A golf course charges \$11.75 one day and gets 50 players. When on another day the rate is reduced to \$11.00, 100 people pay to play, and finally when the rate is \$9.75, 150 people pay to play. What mathematical model would represent this situation?

I approached this problem by figuring that the independent variable is the price, the variation of which causes the variation in the number of players. We then have three sets of points: A (11.75,50), B (11,100) and C (9.75,150). A quick check of the slopes between points A and B and B and C shows that the slope changes, hence the solution would be a non linear expression. I assumed it to be a quadratic equation (with a little bit of cheating, since the author's answer is a quadratic).

Given that p is the price charged and c is the number of players, I figured that the solution to be an expression of the form $ap^2 + bp = c$.

1. For the first set of adjacent points A and B, I then get the simultaneous equations

$$ a*(11.75)^2 + b*11.75 = 50$$ $$a*(11)^2 + b*11 = 100 $$

Similarly the next two points B and C are represented by a set of simultaneous equations as follows: $$ a*(11.00)^2 + b*11.00 = 100$$ $$a*(9.75)^2 + b*9.75 = 150 $$

However I am getting different values for a and b when I solve these two sets of equations.

The author has approached the problem differently, providing an expression for the price in terms of the player count. I am unable to understand the reason for this, given that it is is the price that determines the number of players, and not the player count that determines the price. I would hugely appreciate having someone point out the flaws in my reasoning.

Thanks!

Answer 1

Often, you won't be able to solve 3 equations if you only have 2 variables. If you allow a quadratic of the form: $$a*p^2 + b*p + c = y$$ where y is the number of customers, you may have more luck. Try solving the system of equations: $$a*(11.75)^2 + b*(11.75) +c = 50$$$$a*(11)^2 + b*(11) + c = 100$$$$a*(9.75)^2 + b*(9.75) + c = 150$$ for a, b and c. If you want the answer, according to wolfram alpha, mouseover:

a = -13.33, b = 236.66, c = -890