Extrema of functions of two variables

[Math87]

Homework Statement

A corporation manufactures candles at two locations. The cost of producing x₁ units at location 1 is C₁= 0.02X₁²+4X₁+500
and the cost of producing x₂ units at location 2 is C₂=0.05X₂²+4X₂+275.

The candles sell for $15 dollars per unit. Find the quantity that should be produced at each location to maximize the profit.

p=15(X₁+X₂)-C₁-C₂

The Attempt at a Solution

First off, my professor said "This is just for convenience, especially when plotting it in wolfram alpha, you can't enter x1 and x2 but you can enter x and y. So in the give equations, change the letters, making x1 into x and make x2 into y. Then you will have an equation with numbers and x and y ."
i changed everything to x and y

C₁= 0.02X²+4X+500
C₂= 0.05Y²+4Y+275
Then i decided to combine them together to get:


0.02X²+4X+500-0.05y²-4Y-275=0
After that i found my 6 partial derivatives, but i have a feeling I am doing this all wrong..
Can you give me tips on how to do this question please.

 

[Office_Shredder]

At the end of your post you wrote down the equation C_1 - C_2=0
and then started taking partial derivatives of the left hand side.

This has basically nothing to do with the actual objective: maximizing p(x,y)=15(x+y)-C_1-C_2 = 15(x+y)-.02x^2-4x-500-.05y^2-4y-275

Given a function, how do you find its maximum?

 

[MednataMiza]

I hope I'm not doing this wrong :D
I got few things to say :)
First,you actually can enter values with the name 'x1' and 'x2'. Give it a try :)
Second,in your equation ,where you substituted 'x1' and 'x2' with X and Y.In Mathematica it's better to use lowercase letters for your functions,variables,constants,etc.Some of the uppercase letters are actually preset by the programmers.
Third,the real problem :)
So to find where you will maximize the profit,you should be looking for the lowest budget for producing the candles - the intersection of the two graphics.Finding it by hand it's relatively easy,but you can also use Mathematica.

So the quatity C1=992 and C2=1015,and the profit p(x)=573
Hope I solved that right