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Question 1172441: GetThere Airlines currently charges 200 dollars per ticket and sells 40,000 tickets a week. For every 10 dollars they increase the ticket price, they sell 800 fewer tickets a week. How many dollars should they charge to maximize their total revenue?
I have gotten the answer of $300 by doing these steps. What have I done wrong?
Number of tickets = 40000 - 1000 * (P - 200)/10
Revenue = (40000 - 1000 * (P - 200)/10) * P
= 60000 P - 100 P^2
60000 - 200 P = 0
P = 300
Thanks
Found 2 solutions by math_tutor2020, MathTherapy:
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Currently the revenue is
revenue = (number of tickets sold)*(price per ticket)
revenue = (40,000)*(200)
revenue = 8,000,000 dollars
Let x be the number of times they increase the ticket price by $10
x is some nonnegative whole number.
The old price was 200 dollars
The new price is 200+10x dollars
Eg: if they increase it 3 times, then 200+10x=200+10*3 = 230 dollars is the new ticket price
Each time they increase the price by $10, the amount sold goes down by 800
old amount sold = 40,000
new amount sold = 40,000 - 800x
Eg: if they increased it 3 times, then
40000-800x = 40000-800*3 = 37,600 is the amount of tickets sold.
The new revenue function is therefore
R(x) = (number of tickets sold)*(price per ticket)
R(x) = (40000-800x)*(200+10x)
R(x) = 40000*(200+10x)-800x*(200+10x)
R(x) = 8000000+400000x-160000x-8000x^2
R(x) = -8000x^2 + 240000x + 8000000
The simplified result is of the form
y = ax^2 + bx + c
where
a = -8000
b = 240000
c = 8000000
Use the values of 'a' and b to find the x coordinate of the vertex h
h = -b/(2a)
h = -240000/(2*(-8000))
h = -240000/(-16000)
h = 15
This works because R(x) is a quadratic function that graphs out a parabola. This parabola opens downward producing a highest point at the vertex (h,k). This is where the revenue is maxed out.
A different way to find the x coordinate of the vertex is to locate the roots of R(x). Let's say the roots are p and q. The average of the roots is exactly the value of h. This is due to symmetry. So we can say h = (p+q)/2.
Whichever method you use to find the x coordinate of the vertex, plug that x value into the revenue function to get
R(x) = (40000-800x)*(200+10x)
R(15) = (40000-800*15)*(200+10*15)
R(15) = (40000-12000)*(200+150)
R(15) = (28000)*(350)
R(15) = 9,800,000
The weekly revenue is maxed out at 9.8 million dollars when they increase the ticket price a total of 15 times. Each increase is by $10 increments.
Now the question is what should the ticket price be.
Well that's the result of 200+10x when x = 15, so,
ticket price = 200+10x
ticket price = 200+10*15
ticket price = 200+150
ticket price = 350 dollars
You were fairly close when you got 300 dollars. However, I'm not quite sure how you got those steps.
Answer: 350 dollars
Answer by MathTherapy(10549)  (Show Source):
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