Expert: Ahmed Salami - 11/24/2009

Question
I have started this question but am not sure if I am doing it correctly. Your help will be greatly appreciated. :)

The sum of twice one number and five times a second number is to be 70. What numbers should be selected so that the product of the numbers is as large as possible?

The solution to this problem is 7 and 17.5, but I need to be able to show my work step by step.

This is what I have so far:

70 = 2x + 3y
14 = (2/5)x + y
35 = x + y
y = 35 - x
y' = -1
CN = 0

Is this correct? and what is my next move?

Thank you for your attention to this problem.

Answer
Hi Shawty,
Ok, lets restart.
We have to largest value of the product of two numbers x and y
P = xy
subject to the constraint condition
2x + 5y = 70
from which
5y = 70 - 2x
y = 14 - (2/5)x
Going back to the product equation,
P = x(14 - 2x/5)
 = 14x - 2x²/5
The largest value of P occurs when P' = 0
P' = 14 - 4x/5
when P' = 0
14 - 4x/5 = 0
4x/5 = 14
x = 17.5
and
y = 14 - (2/5)x
 = 14 - (2/5)(17.5)
 = 14 - 7
 = 7

Regards