Maximizing Revenue: Solving for Maximum Profit with the Demand Equation x+9p=450

[Everstar]

Homework Statement

The Demand Equation for a certian product can be modeled by x+9p=450. Use this to rewrite the revenue equation R=xp as a function of x. Find the number of goods that will maximize profit. Find the maximum profit.

Homework Equations

x+9p=450
R=xp

The Attempt at a Solution

I have no idea =S. My real problem is i don't know how to write r=xp as a function of x i know it will end up being a quadratic equation i just don't know how to do it. Thanks so much for any help.

 

[Everstar]

Like in my math book examples it takes x=21,000-150p and turns it into R=xp=(21,000-150p)p=-150p^2+21,000p. It says that's expressing the revenue r as a function of p. I don't even understand what they did and that's the first example. I miss 2 days and i get so far behind =(

 

[Char. Limit]

Well, first you need to isolate p in your first equation. By this I mean manipulate the equation until you have p, alone, on one side. Then you should have p as a function of x. After this, you can say that p is equal to this function of x, and replace p in your second equation with the function of x.

 

[Everstar]

so i get p=50-x/9?

Im sorry but i don't understand your second part.

 

[Everstar]

oh wait so r=x(50-x/9) right?

 

[Char. Limit]

Exactly.

Now factor all of that out, and you get your quadratic.

 

[Everstar]

R=50x-x^2/9 is what i got
well can't i like write it like this R(x)=-1/9x^2+50x

1/9 is the same as dividing by 9?

 

[Char. Limit]

Sounds good.

 

[Everstar]

alright cool thank you!

 

[Everstar]

Now to find the number of good that will maximize the profit i take h=-b/2a witch is -50/2(-1/9)= 225.

I put the 225 in the equation for x like: R(225)=-1/9(225)^2+50(225)= 5626 and that's my maxium profit correct?

 

[Char. Limit]

Now to find the number of good that will maximize the profit i take h=-b/2a witch is -50/2(-1/9)= 225.

I put the 225 in the equation for x like: R(225)=-1/9(225)^2+50(225)= 5626 and that's my maxium profit correct?

I think there's a typo there, since your math and the correct answer both total to 5625. Maybe you intended to hit that instead?

If you did, then yes, that is your maximum profit.

 

[Everstar]

ok haha yeah i did thanks for all the help =)