# Thread: Optimization problem (finding 2 numbers given an criteria)

1. ## 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. $\displaystyle x-y=54$ seems simple enough, but I don't know what to do for the second.

2. $\displaystyle x-y=54 \implies x=54+y$

Now

$\displaystyle x^2\times 6y =$ min

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

Solve for

$\displaystyle ((54+y)^2\times 6y)' =0$

3. 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. $\displaystyle x-y=54$ seems simple enough, but I don't know what to do for the second.
$\displaystyle x - y = 54 \, \implies \, y = x-54$

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

minimize the product, $\displaystyle P$