
Rectangular Packing Box
A rectangular packing box, similar to a shoebox, without a lid, is 3 times as long as it is wide and half as high as it is long. Each square inch of the bottom of the box costs $0.008 to produce, while each square inch of any side costs $0.003 to produce.
(A) Write an equation that represents the cost, c, of the box as a function of its width x.
(B) Using the function written in part a, determine the dimensions of a box that would cost $0.69 to produce.

Hello,
so its width is x.
We know that its length is 3 times as long as its width, meaning that its length is 3x.
We know that its height is half as long as its length, meaning that its height is (3x)/2.
If you view the box (I recommend you make a sketch), there is one bottom, rectangular, with its sides : length,width,length,width.
So its area is (width * length) = 3x²
For two of the sides, the sides are : length,height,length,height.
So each of their area is (length * height) = 9x²/2
For the last two sides, the sides are : width,height,width,height.
So its area is (width * height) = 3x²/2
And finally, the cost will be ... ?
For question b), solve for x in f(x)=0.69.
This is not hard, just taking square roots ;)
If you struggle with that, let us know.