optimization problems

[mike1822]

Hi I cant solve these two optimization problems...


1. suppose that 20,000 fans will go to a ball game when the price of a ticket is $5, and that 500 fewer fans will go each $1 increase in price. What should the ticket price be in order to maximize revenue?



2. compute the max profit when the demand function is p(x)=x^2-3x+2 and the total cost function is c(x)=2x^3/3-1/2x^2-2x. Recall that r(x) = xp. Enter just a reduced fraction of form a/b.


Can anyone solve these?

Thanks

 

[mike1822]

the first one I set up in y-y1 = m(x-x1) form. with the points being (5,2000) and (6,19500). It came out to be y=-500x+22500. I then did R(x)=xp
R(x)=x(-500x+22500) and distributed the x. Then i took the first derivitve and solved for 0. I got 22.5 but I think I solved for something else because that price is too high.

 

[mike1822]

As far as the second one I tried just plugging the two formulas in the formula p(x) = R(x)-c(x), taking the first derivitive and then solving for 0, but Im lost on it and really dont know what to do.

 

[BigGlenntheHeavy]

\(\displaystyle 1) \ R(n) \ = \ -500n^{2}+17500n+100000\)

\(\displaystyle R'(n) \ = \ -1000n+17500 \ = \ 0, \ n \ = \ \frac{35}{2} \ = \ 17.5\)

\(\displaystyle Hence, \ 22.50 \ a \ ticket \ will \ maximize \ revenue.\)

\(\displaystyle Jam \ it \ to \ the \ suckers.\)