Question

Calculus Help!
A company produces x units of commodity A and y units of commodity B each hour. The company can sell all of its units when commodity A sells for p=25−5x dollars per unit and commodity B sells for q=45−10y dollars per unit. The cost (in dollars) of producing these units is given by the joint-cost function C(x,y)=5xy+3. How much of commodity A and commodity B should be sold in order to maximize profit?

Answer 1

Have you made any progress on this problem thus far?

Answer 2

no

Answer 3

The first objective would be finding the profit function to maximize. This is expressed as the revenue from selling subtracting the cost for production. We know our prices for commodity A and B per unit so the revenue can be determined by multiplying units sold (x or y) by its price per unit (p or q). $/unit * unit = $

Answer 4

So the equation would be (5xy+3)*(25-5x)?

Answer 5

Well, we sell x units of commodity A for p=25-5x dollars each. That means the total amount we make is just x*p or x*(25 - 5x). The same occurs for commodity B with y and q. The total revenue, then, is what we make total. x*p + y*q = revenue. Can you see where I get that?

Then we know costs, so profit = revenue - costs.

Answer 6

Ok yeah.
so (25x-(5x^2))+(45y-(10y^2))-(5xy+3)

Answer 7

Yes! :)
So then our goal next is to maximize that function to find optimum values of (x, y). Do you know that process from here?

Answer 8

Take the partial derivative of x and y?

Answer 9

Sounds good to me.

Answer 10

ok so
P(x)=25-10x-5y
P(y)=45-20y-5x

Answer 11

The partial derivatives are correct to me.

Answer 12

and then if we want to find a critical point we:
P(y)=(45-20y-5x)*-2=(-90+10x+40y)
P(x)=(25-10x-5y)

Which equals:
-65+35y=>y=65/35
?

Answer 13

Yes, that looks good. :)
I made a small error myself where I must have wrote q=40-10y instead of 45-10y. But I fixed it and the answers match again!

Answer 14

Cool!
So than I know I have to plug in (65/35) for y but I don't know which "y" to plug it in to lol

Answer 15

Both should work to be the same, since it is essentially the intersection point of the two linear equations.

Answer 16

45 - 20y - 5x = 0 <-- y=65/35
or
25 - 10x - 5y = 0 <-- y= 65/35

in either case, x will be the same.

Answer 17

Alright so 25-10x-5(65/35)=approx 1.571428517!!!

Answer 18

Looks good to me! Could also be represented in fractions for the nicer numbers: x=11/7, y=13/7.

I am thinking that is the answer, although I don't know if the problem wants to assume x and y have to be integers (like, you can't sell partial commodity). I might be overcomplicating that. lol

Answer 19

Yes it is the right answer (the homework is online so we know instantly if it's right)!

Answer 20

Alright, great! :P

Answer 21

Thank you,
So one final quick question... Would I do about the same thing for:


"A manufacturer is planning to sell a new product at the price of $290 per unit and estimates that if x thousand dollars is spent on development and y thousand dollars is spent on promotion, consumers will buy approximately (250y/y+4)+(500x/x+3) units of the product. If manufacturing costs $120 per unit, how much should the manufacturer spend on development and how much on promotion to generate the largest possible profit from the sale of this product?"

Answer 22

Your goal of the equation "profit = revenue - cost" is the same, it just looks like you have to find revenue and cost differently from the first one.

Answer 23

So,
P would=1700((25y)/y+4)+((50x)/x+3)-1000x-1000y

Answer 24

Is that written like:
\[ 1700 \frac{25y}{y + 4} \cdots \]
I think it looks correct from that.

Answer 25

Ok so then the partial derivative would be?

Answer 26

There is a quotient of two polynomials, so you would use quotient rule for derivatives to find that part.

Answer 27

would P(x)=-(1000 (-246+6 x+x^2))/(3+x)^2
would P(y)=-(1000 (-154+8 y+y^2))/(4+y)^2?

Answer 28

Sorry, I wasn't getting that for an answer. I need to retry my steps...

Answer 29

Ok no problem, If you can't get it it's no problem because my homework was due at 11 lol. Thanks for all the help though!

Answer 30

I think you are correct, I just seem to be getting a weird answer. Might be the sleepiness. lol

Answer 31

Alright I'll let you go. Again, thanks for the help!

Answer 32

No problem! I think once you got the function for profit, you are fine! :D

Answer 33

I glanced over my work and found I had written 50 instead of 150. Fixing that, I got the same partial derivatives you had. lol