Question 1002286: A city's transit authority serves 192,000 commuters daily when fair is $1.70. Market research has determined that every penny decrease in the fare will result in 1200 new riders. What fare will maximize revenue?
We are working on Quadratic Functions, and word problems are especially difficult for me to figure out. This is what I have and the answer is incorrect:
Revenue = Price(number sold)
Revenue = y
y=P(1.70-P)
y= 1.70p-p^2
using the formula for a Vertex of a Parabola: (-b/2a, c-b^2/4a)
a= -1 b= 1.70 c= 0
using the "x" part of that formula:
x= -1.70/2(-1)
I got the answer of $0.85 (WHICH IS INCORRECT)
Thank you
Angy
Found 3 solutions by vleith, Theo, KMST:
Answer by vleith(2983)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! this one is particularly difficult to set up so don't feel too bad.
i haven't done it in a while and i'm struggling to set it up properly.
start with r = n * p
r = revenue
n = number of tickets sold
p = price per ticket.
you know that, when p = 1.71, n = 192000
you get r = 192000 * 1.71 = 328320
you also know that, when you drop the price by .01, n goes up 1200.
let 192000 + 1200 * x = the increase in n for every drop in price of .01
let 1.71 - .01 * x = the drop in price of .01.
every time x increases by 1, the price will drop by .01
every time x increases by 1, the number of tickets sold will go up 1200.
your formula is:
y = (192000 + 1200 * x) * (1.71 - .01 * x)
y is the revenue
(192000 + 1200 * x) is the number of tickets sold.
(1.71 - .01 * x) is the price of each ticket.
if x is 0, then you get:
y = (192000 + 1200 * x) * (1.71 - .01 * x) becomes:
y = (192000 + 1200 * 0) * (1.71 - .01 * 0) which becomes:
y = 192000 * 1.71 = 328320
any time x increases by .01, the number of tickets sold increases by 1200.
this is because the price has decreased by .01.
any time x decreases by .01, the number of tickets sold decreases by 1200.
this is because the price has increased by .01.
your maximum point will be when x = 5.5
you will get:
y = (192000 + 1200 * 5.5) (1.71 - .01 * 5.5) which results in:
y = 328683.
that's your maximum revenue.
simplify the equation by multiplying the factors to get:
y = (192000 + 1200 * x) * (1.71 - .01 * x) becomes:
y = (192000 * 1.71) - (192000 * .01 * x) + (1200 * x * 1.71) - (1200 * x * .01 * x)
simplify to get:
y = 328320 - (1920 * x) + 2052 * x) - (12 * x^2)
combine like terms to get:
y = 328320 + 132 * x - 12 * x^2
re-order terms in descending order of degree to get:
y = -12x^2 + 132x + 328320
now that your equation is in standard quadratic form, you can solve for maximum point as follows:
a = -12
b = 132
c = 328320
x = -b/2a becomes x = -132 / -24 which becomes x = 5.5
when x = 5.5, the equation of y = -12x^2 + 132x + 328320 becomes:
y = -12(5.5)^2 + 132(5.5) + 328320 which becomes:
y = 328683
your maximum revenue is at 328683
this occurs when x = 5.5
when x = 5.5, your price becomes 1.71 - .01 * 5.5 which becomes 1.71 - .055 which becomes 1.655.
your maximum revenue is when the price is equal to 1.655.
this can also be found by graphing the equation of y = -12x^2 + 132x + 328320 as shown below:
Answer by KMST(5328)  (Show Source):
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