Word Problem: find ticket price yielding greatest revenue

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iRishad

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Need some help with this word problem. You don't have to give me the answer, just show me the steps and show your work, mathematically.

1) The cost of a ticket to a hockey arena searting 800 people is $3. At this price every ticket is sold. A survey indicates that if the price is increased, attendance will fall by 100 for every dollar of increase. What ticket price results in the greatest revenue. What is the greatest revenue.

I tried this question a million times. I did this by the Revenue equation(Revenue = Price x Quantity). My equation was like this:

(3 + x)(800 -100x) ...solved that into 2400 - 300x + 800x - 100x2 or 2400 + 500x - 100x2

After that ... I divided 500x by 2(to get the optimum value).

Anyways, I'm pretty sure I'm wrong...so any help would be good.
 
Re: Word Problem

iRishad said:
Need some help with this word problem. You don't have to give me the answer, just show me the steps and show your work, mathematically.

1) The cost of a ticket to a hockey arena seating 800 people is $3. At this price every ticket is sold. A survey indicates that if the price is increased, attendance will fall by 100 for every dollar of increase. What ticket price results in the greatest revenue. What is the greatest revenue.

I tried this question a million times. I did this by the Revenue equation(Revenue = Price x Quantity). My equation was like this:

(3 + x)(800 -100x) ...solved that into 2400 - 300x + 800x - 100x2 or 2400 + 500x - 100x2

everything is fine up to this point ... find the x-value of the vertex, x = -b/(2a), to determine the x-value that will maximize R = 2400 + 500x - 100[sup:3lue5jsf]2[/sup:3lue5jsf] ... once you've determined that value of x, then evaluate R with that value to find the max revenue.
Click to expand...
 
iRishad said:
Need some help with this word problem. You don't have to give me the answer, just show me the steps and show your work, mathematically.

1) The cost of a ticket to a hockey arena searting 800 people is $3. At this price every ticket is sold. A survey indicates that if the price is increased, attendance will fall by 100 for every dollar of increase. What ticket price results in the greatest revenue. What is the greatest revenue.

I tried this question a million times. I did this by the Revenue equation(Revenue = Price x Quantity). My equation was like this:

(3 + x)(800 -100x) ...solved that into 2400 - 300x + 800x - 100x2 or 2400 + 500x - 100x2

After that ... I divided 500x by 2(to get the optimum value).

Anyways, I'm pretty sure I'm wrong...so any help would be good.
Click to expand...


Revenue = number of tickets sold * cost of 1 ticket

When the price is $3, all 800 tickets are sold, bringing in a revenue of $2400.

Let x = number of $1 increases in the cost of a ticket

We are told that for every $1 increase in cost, there are 100 fewer attendees. So, if you make "x" increases of $1, the number of attendees will be

800 - 100x

Revenue when the cost of a ticket is $3 + x would be (800 - 100x)*(3 + x)

Let y = revenue.

y = (800 - 100x)(3 + x)
or,
y = 100(8 - x)(3 + x)

You've got an equation in the form

y = a(x - b)(x - c)

which is known as the "intercept form" for the equation of a parabola.

The x-intercept occurs

at x - b = 0, and x - c = 0....or, in your specific case, when 8 - x = 0 or when 3 + x = 0.

Ok....the vertex of the parabola (which is the maximum or minimum value of the function) occurs halfway between the x-intercepts.

I hope you can find the vertex, and thus the max or minimum value for the function.

If you are still having trouble, please repost, and show us all of the work you've done to solve this problem.
 
Mrspi,

I understand everything up till:

y = 100(8 - x)(3 + x)

^^Doesn't that have to be in the structure --> y = a(x - s)(x - t)

So shouldn't it be y = 100(x - 8)(x + 3)

But then it messes everything up ---> (8 + -3) / 2 = 5 (not 11 :( )
 
\(\displaystyle (3+x)(800-100x)=x^{2}-5x-24\)

\(\displaystyle \frac{-b}{2a}=\frac{5}{2}=2.5\)
 
Don't think that's right ^

The ticket price is $5.5 , just trying to figure out how it is.
 
iRishad said:
Don't think that's right ^

The ticket price is $5.5 , just trying to figure out how it is.
Click to expand...

(3 + x)(800 -100x)
= 800(3 + x) -100x(3 + x)
= -100x^2 + 500x + 2400
= x^2 -5x -24

-b/(2a) = 5/(2) = 2.5
(and, of course: 2x - 5 = 0, x = 5/2 = $2.5)

Yup..... that is correct!
 
But how ?

$2 = 900 = 1800
$2.5 = 850 = 2125
$3 = 800 = 2400
$4 = 700 = 2800
$5 = 600 = 3000
$5.5 = 550 = 3025
$5 = 500 = 2500

How is it $2.5 ?
 
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