SOLUTION: All day I've tried to get this word problem completed and I have no clue. Can you help me? An open-top box is to be constructed from a 6 foot by 8 foot rectangular cardboard by

Question 83177: All day I've tried to get this word problem completed and I have no clue. Can you help me?
An open-top box is to be constructed from a 6 foot by 8 foot rectangular cardboard by cutting out equal squares at each corner and folding up the flaps. Let x denote the length of each side of the square to be cut out.
Questions:
Find the function V that represents the volume of the box in terms of x.
Graph this function and show the graph over the valid range of the variable x.
Using the graph, what is the value of x that will produce the maximum volume?
Found 2 solutions by Mona27, jim_thompson5910:
Answer by Mona27(45) (Show Source):
You can put this solution on YOUR website!
If you can imagine the cardboard 6 feet by 8 feet, and you're cutting squares of side x from each corner.
The length of the box would be 8-2x, and the width would be 6-2x, and the height is x.
So the volume is
The relevant range of values of x are from 0 to 3, since you can't have the cut-out squares having a larger side than half the width of the original rectangular cardboard..
The graph looks like this:

And the value of x which will give the maximum volume is approximately 1.13

Answer by jim_thompson5910(21685) (Show Source):
You can put this solution on YOUR website!
a)
It's best to draw a picture for this one. So just draw a rectangle with squares of side length of x cut out of the corner and with dimensions of (8-2x) and (6-2x).

The rectangle left inside will be the base with sides of and (its 2x taken away since there are 2 sides of 2 corners per side) and the outer rectangles will form the vertical walls of the box which means the box will have a height of x. I hope this picture is starting to make sense.
This means that the area of the base is

And since the height is x. So the volume is

b)
Graph of x(8-2x)(6-2x)
The domain of x that makes sense is the values that produce a positive y (negative volume doesn't make sense) and x is between 0 and 3. Anything over x=3 means there is a negative value associated with the volume which doesn't make sense.

c)
Continuing from b) our attention is focused on the first peak, it turns out that the max volume is the apex of the curve (in other words the highest point in the range of x=0 to x=3). If you graphed and found the max with your calculator it would be (1.131,24.258) So that means the max volume you could get would be about 24.25 cubic feet with the x cutout of 1.131 feet.
Hope this helps. It really helps to draw the rectangle with the square corner cutouts and everything labeled.