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Nathan4102

Sun May 12, 2013 5:26 pm


Another simple quadratic question!!!

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Alright, so I'm working on more of this quadratics work, and I've come across another problem. Its not that I can't solve this one, its that I think I'm doing it the wrong way. It took me about 3 minutes to find the equations for the let statements, and a decent amount of time to solve the equation because of all the fractions. Obviously, I need all the time I can get on tests and exams, so is there a cleaner, better, and faster method to solve this?

Q: A bus company carries about 20 000 riders per day for a fare of $0.90. A survey indicates that if the fare is decreased, the number of riders will increase by 2000 for every decrease of $0.05. Find the ticket price that will result in the greatest revenue.

A:
[CODE]
Note: 56 000 - 40 000x = 20 000 + 40 000(.9 - x)

let x rep the bus fare.
let 56 000 - 40 000x rep the amount of riders riding the bus.
let y rep the total daily revenue.
y = (x) (56 000 - 40 000x)
y = 56 000x - 40 000x^2
y = -40 000 (x^2 - 7/5x)
y = - 40 000 (x^2 - 7/5x + 49/100) - 49/4 000 000
y = -40 000 (x - 7/10)^2 - 49/4 000 000

7/10 = 0.7

Therefore the ticket price should be $0.70 in order to achieve maximum daily revenue.
[/CODE]


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whoareyou

Sun May 12, 2013 6:06 pm


RE:Another simple quadratic question!!!

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When you're doing quadratic optimization, most of the time you'll need to find the vertex of the function. The x-value of the function is -b/2a when the equation is in the form ax^2 + bx + c. So at the second line after your let statements, -56000/-80000 = 0.7.


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TZak

Sun May 12, 2013 6:06 pm


Re: Another simple quadratic question!!!

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Find the derivative.


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whoareyou

Sun May 12, 2013 6:08 pm


RE:Another simple quadratic question!!!

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Sorry, "The x-value of the function is" should read "The x-value of the vertex is."


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Nathan4102

Sun May 12, 2013 6:11 pm


Re: RE:Another simple quadratic question!!!

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When you're doing quadratic optimization, most of the time you'll need to find the vertex of the function. The x-value of the function is -b/2a when the equation is in the form ax^2 + bx + c. So at the second line after your let statements, -56000/-80000 = 0.7.
"-b/2a"? Not too sure what you mean by this. This is only grade 10, btw, in case thats 11+.


Find the derivative.


We haven't been taught derivatives yet. I could learn them on my own, but my teacher wouldn't mark it.


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Nathan4102

Sun May 12, 2013 6:12 pm


Re: RE:Another simple quadratic question!!!

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Sorry, "The x-value of the function is" should read "The x-value of the vertex is."

Ok, now the -b/2a makes a bit more sense, but could you explain how you got it?


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Zren

Sun May 12, 2013 6:14 pm


RE:Another simple quadratic question!!!

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Where did 56 000 come from?

I interpreted it as:

Revenue = Riders * Cost
Revenue = 20000 * $0.90

let y represent revenue
let x represent ______