SOLUTION: I have a word problem. "A tour bus serving 400 customers a day charges $5 per person. The owner estimates that the company will lose 20 customers for every $3 fare increase. Wha

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons -> SOLUTION: I have a word problem. "A tour bus serving 400 customers a day charges $5 per person. The owner estimates that the company will lose 20 customers for every $3 fare increase. Wha      Log On


   

Question 714917: I have a word problem.
"A tour bus serving 400 customers a day charges $5 per person. The owner estimates that the company will lose 20 customers for every $3 fare increase. What charge would you give the most income for the company and what daily income would they expect."
I think the setup is like this, but I am not sure: (400-20x)(5+3x).
COULD YOU PLEASE SHOW THE STEPS? THANKS A LOT.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
If x is the number of three dollar fare increases and y is the income, then
y+=+%28400-20x%29%285%2B3x%29
would be the equation we could use to solve this problem. (Note: Figuring out (400-20x)(5+3x) was probably the hardest part of this problem.)

The solution to the problem will become more apparent if we multiply out the right side (using FOIL):
y+=+400%2A5+%2B+400%2A3x+%2B+%28-20x%29%2A5%2B%28-20x%29%2A3x%29
Simplifying...
y+=+2000+%2B+1200x+%2B+%28-100x%29%2B%28-60x%5E2%29
y+=+2000+%2B+1100x+%2B%28-60x%5E2%29
We can now recognize that the equation is a quadratic equation. Putting this equation in standard ax%5E2%2Bbx%2Bc form we get:
y+=+%28-60x%5E2%29+%2B+1100x+%2B2000

We should know that the graph of all quadratic equations is a parabola. And with an x%5E2 term this parabola will open upwards or downwards. And with a negative coefficient in front of the x%5E2 this parabola will open downwards.

If you imagine (or draw) such a parabola you should recognize that the highest point of such a parabola will be its vertex. And since the y coordinate of the points on this parabola represents the income, the maximum income will occur at the vertex.

So the problem boils down to finding the coordinates of the vertex. This can be done...
  • By completing the square and putting the equation into vertex form:
    4p%28y-k%29+=+%28x-h%29%5E2
    where the coordinates of the vertex, (h, k), can be simply read from the equation; or
  • By ...
    1. memorizing and using the fact that the x-coordinate of the of the vertex of a parabola in standard form will always be:
      %28-b%29%2F2a
    2. Using the x-coordinate of the vertex and the equation of the vertex to find the y-coordinate of the vertex.
Since completing squares is a lot of work, I'm going to be lazy and use the second approach.

In our equation, b = 1100 and a = -60. So the x-coordinate of the vertex will be:
%28-1100%29%2F%282%2A%28-60%29%29
Simplifying...
%28-1100%29%2F%28-120%29%29
110%2F12
Since x is the number of three dollar fare increases, it must be a whole number. The closest whole number is 9. (We'll also try the whole number on the "other side", 10, just to make sure 9 gives us the maximum income.)
Substituting 9 into the equation we get:
y+=+%28-60%289%29%5E2%29+%2B+1100%289%29+%2B2000
Simplifying...
y+=+%28-60%2A81%29+%2B+1100%289%29+%2B2000
y+=+%28-4860%29+%2B+9900+%2B2000
+y+=+7040

Just to make sure that this is the max we will try the other whole number near 110/12: 10
y+=+%28-60%2810%29%5E2%29+%2B+1100%2810%29+%2B2000
Simplifying...
y+=+%28-60%2A100%29+%2B+1100%2810%29+%2B2000
y+=+%28-6000%29+%2B+11000+%2B2000
+y+=+7000

So we get the maximum income when x = 9. Since x represents the number of three dollar fare increases above $5, the fare that results in the maximum income will be:
5 + 3*9 = 5 + 27 = 32
And the income this fare will generate will be $7040.