Find two positive numbers such that their product is 192 and their sum is a minimum pre-calc

This problem is a combination of algebra and calculus. First, we formulate an equation for the product. Let x be the first positive number and y be the other positive number. So, the equation would be

xy = 192

Let's rearrange this to make the equation explicit: y = 192/x

Then, the other equation is for the sum. Let S be the sum of the two positive numbers.

S = x + y

Now, substitute the other equation into this one:

S = x + 192/x

Here's where we apply calculus. We can determine the minimum S by getting the first derivative of S with respect to y and equating to zero. Thus,

dS/dx = 1 - 192x⁻² = 0

1 = 192x⁻²

x² = 192

x = + / - √192

But since we are finding the positive number, x = + √192

Then, we use this to the first equation:

y = 192/√192 = √192

Therefore, the two positive numbers are equal which is √192 or 13.86.