# Optimization problem (finding 2 numbers given an criteria)

• Apr 4th 2011, 02:55 PM
IanCarney
Optimization problem (finding 2 numbers given an criteria)
Find two numbers with a difference of 54 such that the square of one number multiplied by 6 times the other is a minimum.

I'm not sure how I would go about making a relevant equation. $x-y=54$ seems simple enough, but I don't know what to do for the second.
• Apr 4th 2011, 03:26 PM
pickslides
$x-y=54 \implies x=54+y$

Now

$x^2\times 6y =$ min

$(54+y)^2\times 6y =$ min

Solve for

$((54+y)^2\times 6y)' =0$
• Apr 4th 2011, 03:26 PM
skeeter
Quote:

Originally Posted by IanCarney
Find two numbers with a difference of 54 such that the square of one number multiplied by 6 times the other is a minimum.

I'm not sure how I would go about making a relevant equation. $x-y=54$ seems simple enough, but I don't know what to do for the second.

$x - y = 54 \, \implies \, y = x-54$

$P = x^2\cdot 6(x - 54)$

minimize the product, $P$