how do I find the maximum values?

May 2010
22
0
Mr. MacGregor's Shop makes bookcases and desks. Each bookcase requires 5 hours of woodworking and 4 hours of finishing. Each desk requires 10 hours of woodworking and 3 hours of finishing. Each month the shop has 600 hours of labor available for woodworking and 250 hours for the finishing. The profit on each bookcase is $40 and each des is $75. How many of each product should be made each month in order to maximize to profit?
 
May 2010
1,260
410
Mauritius
I'm not too sure about it... but that's what I would have done:

For maximising profit, I take that MacGregor needs to make maximum use of all his given time.

Total time for woodworking is:

600 = 5b + 10d

where b and d are the number of bookcases and desks respectively.

Total time for finishing is:

250 = 4b + 3d

By solving these two simultaneous equations, I find that:
b = 28
d = 46

This makes a profit of $ 4710

Now, if he wants to make more desks, he'll need to make 47 desks and 26 bookcases so that the time constraints are still respected.

5(26) + 10(47) = 600 (good)
4(26) + 3(47) = 245 (good, still less than 250)
Profit = $ 4565 (ok, it's less, so, my answer is still the more profitable)

Now, if he wants to make more bookcases, he'll need to make 44 desks and 29 bookcases so that the time constraints are still respected.

5(29) + 10(44) = 585 (good)
4(29) + 3(44) = 248 (good)
Profit = $ 4460 (ok, it's less, so, my answer is still the more profitable)

My conclusion, the number of bookcases and desks to make maximum profit is 26 and 46 respectively :)