Question

Find two positive real numbers such that the sum of the first number squared and the second number is 147 and their product is a maximum.

Answer 1

If you give something a name, you get control over it. So it says you have two positive real numbers, let's give these numbers names, r and s. No particular reason why I named them r and s, just liked 'em.

So now the problem says that the sum of the first number squared, ( that means \(r^2\) right? ) and the second number, \(s\) is 147. So the sum of \(r^2\) and \(s\) is 147. We can say this exact same thing, in an even shorter way as: \(r^2+s=147\).

There's one last bit, their product must be maximum. Alright, \(r*s\) needs to be maximized. Let's go ahead and call this thing \(p=r*s\) since \(p\) is the first letter of product and it's what we wanna maximize. How can we maximize this? Well, if you solve for one of the variables in our constraint, \(r^2+s=147\) and we plug it into our formula for p, we can then take its derivative and set it equal to 0 since a maxima of a function happens at the top where it's flat.

Give it your best shot.