find two numbers whose difference is 22 and whose product is as small as possible

Question

zepdrix · Answer

Hey Ivan :) Welcome to OpenStudy!

What type of class is this for?
Can we use calculus techniques?

anonymous · Answer

college algebra, and yes any technique works...

zepdrix · Answer

Ooo boy :( I'm trying to remember how to do optimization problems without using Calculus. hmmm

zepdrix · Answer

Oh oh oh ok ya ya ya, we can do this.

zepdrix · Answer

The difference of two numbers is 22.$$\Large\rm x-y=22$$Their product is a minimum.$$\Large\rm P_{\min}=xy$$I'm calling it P for product.

If we use our first equation,
add y to each side,

zepdrix · Answer

$$\Large\rm \color{orangered}{x=22+y}$$

zepdrix · Answer

We can plug this into our product equation:$$\Large\rm P_{\min}=\color{orangered}{x}y$$$$\Large\rm P_{\min}=\color{orangered}{(22+y)}y$$

zepdrix · Answer

Distribute,$$\Large\rm P_{\min}=y^2+22y$$So what we have here is a parabola.
You have to remember a little something about parabolas in order to solve this equation.
The leading term is positive, so it's opening "up".
So our minimum is going to be the `vertex` of this parabola.

zepdrix · Answer

So we need it in vertex form.

anonymous · Answer

difference is 22 and product is as mall as possible?