SOLUTION: 1) 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 the folding up the flaps. Let x denote the le

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Question 73411: 1) 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 the folding up the flaps. Let x denote the length of each side of the square to be cut out.
a) Find the function V that represents the volume of the box in terms of x.
Answer



b) Graph this function and show the graph over the valid range of the variable x..
Show Graph here



c) Using the graph, what is the value of x that will produce the maximum volume?
Answer
Found 2 solutions by jim_thompson5910, ankor@dixie-net.com:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a)
It's best to draw a picture for this one, but I don't know how to draw here. So just draw a rectangle with squares cut out of the corner. The rectangle left inside will be the base with sides of 8-2x and 6-2x (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
%288-2x%29%286-2x%29
And since the height is x. So the volume is
V=base%2Aheight%2Adepth=x%288-2x%29%286-2x%29

b)
graph%28+300%2C+200%2C+-2%2C+8%2C+-2%2C+25%2C+x%2A%288-2x%29%2A%286-2x%29%29+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 x%288-2x%29%286-2x%29 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.

Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
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 the folding up the flaps. Let x denote the length of each side of the square to be cut out.
:
A few things we know
The length = (8 - 2x)
The width: = (6 - 2x)
The height = x
:
a) Find the function V that represents the volume of the box in terms of x.
Answer
V(x) = (8-2x) * (6-2x) * x
:
FOIL
V(x) = (48 - 16x - 12x + 4x^2) * x
:
V(x) = x(4x^2 - 28x + 48)
:
V(x) = 4x^3 - 28x^2 + 48x
:
:
b) Graph this function and show the graph over the valid range of the variable x..
Show Graph here
+graph%28+300%2C+200%2C+-1%2C+3%2C+-10%2C+30%2C+4x%5E3+-+28x%5E2+%2B+48x%29+
:
When you plot this. Plot every 1/4 ft: x = .25, .5, .75. 1.00,
Then plot every 1/10 ft: 1.1, 1.2, 1.3 1.4, 1.5,
Then plot every 1/4 ft: 1.75, 2.00 etc
This will give you a close value for the max.
:
c) Using the graph, what is the value of x that will produce the maximum volume?
:
Answer: It looks like x is slightly greater than 1 ft, say 1.1 ft and the actual volume is about 24 cu ft