Hello M7MBA.

Let's take a look at your question. We have the next rectangle:



The area of the rectangle is 18, that is to say, $$x\cdot y=18.$$. Also, "the average length of all the sides of a rectangle equals twice the width of the rectangle" means $$\frac{x+y+x+y}{4}=\frac{2x+2y}{4}=\frac{x+y}{2}=2\cdot y.$$

We can rewrite this last equation as follows: $$x+y=4y\ <-->\ x=3y.$$

If we replace this last equation in the area equation we will get: $$x\cdot y=18<-->\ 3y\cdot y=18<-->y^2=6<-->\ y=\sqrt{6}.$$

This implies that $$x=3\sqrt{6}.$$ and the perimeter is equal to $$2x+2y=6\sqrt{6}+2\sqrt{6}=8\sqrt{6}.$$

So, the correct answer is B.

I hope this can help you to understand the calculus.

I'm available if you'd like a follow up.

Regards.