# Maximizing Profit question

• Mar 28th 2009, 03:09 PM
zanyspydude
Maximizing Profit question
This one is giving me some trouble. Can you help me find the solution?

I will be paid 5$for every every book I bring to a man. I will be paid an extra$1 per book per box if I bring them in a box.

It will take me 300 minutes to find 1000 books. It will take me 500 minutes to find 1 box.

I have a given number of minutes to return. What is the best strategy to take in order to earn the most?

What I can figure out so far:

let x be the number of books and y be the number of boxes.

Profit = x * ($5 +$1 * y)
Time = 300(x/1000) + 500y

How do I put the two of these together to figure out the max?
• Mar 28th 2009, 04:49 PM
zanyspydude
My attempt at lagrange has gotten me no where:

f(x,y) = x * (5+y)
g(x,y) = 300(x/1000) + 500y

F(x,y) = f(x,y) - lambda(g(x,y) - k)
F(x, y, lambda) = x * (5 + y) - lambda(300/1000x + 500y - k)
F(x, y, lambda) = 5x + xy - 3/10 lambda x - 500 lambda y + lambda k

Fx = 5 + y - 3/10 lambda - 500y = 0
Fy = x - 500 lambda = 0
Flambda = -3/10x - 500y + k = 0
• Mar 28th 2009, 06:30 PM
Ond
Are the minutes you have to do this not given in the example?
• Mar 29th 2009, 09:10 AM
zanyspydude
Quote:

Originally Posted by Ond
Are the minutes you have to do this not given in the example?

They are but I'd like to solve for the general case of k. K= 65000
• Mar 29th 2009, 10:24 AM
Ond
Don't know if this is what you're looking for, but here it goes...

We have the following:
1) $\displaystyle F_x=5+y-0,3\lambda-500y=0$
2) $\displaystyle F_y=x-500\lambda=0$
3) $\displaystyle F_\lambda=-0,3x-500y+k=0$

From 2) we have:
$\displaystyle \lambda=x/500$

This into 1) and solve for x gives us:
$\displaystyle 5+y=0,3x/500+500y$
$\displaystyle x=8.333,33+1.667,67y$

We then stick this expression in equation 3) above and solve for y:
$\displaystyle k=0,3(8.333,33+1.667,67y)+500y$
$\displaystyle k=2.500+500y+500y$
$\displaystyle 1.000y=k-2.500$
$\displaystyle y=k/1.000-2,5$

Above where it says "This into 1)..." you just solve for y instead of x and then do the same steps. That will then yield x in terms of k.

Hopefully my calculations are right here, and hopefully this is what you were looking for?