SOLUTION: An open top box is to be constructed from a 6 foot by 8 foot rectanglar cardboard by cutting out equal squares at each corner and the folding up the flaps. Let x denote the length

Question 73932: An open top box is to be constructed from a 6 foot by 8 foot rectanglar 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
b) graph the function and show the graph over the valid range of the variable x
c) Using the graph, what is the value of x that will produce the maximum volume?
Found 2 solutions by ankor@dixie-net.com, stanbon:
Answer by ankor@dixie-net.com(22740) (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

:
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

Answer by stanbon(75887) (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 rectanglar 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.
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Draw the picture so you can see the width, length, and height of the box.
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a)find the function V that represents the volume of the box in terms of x
New width is 6-2x ft , New length is 8-2x ft ; height = x ft.
Volume = width*length*height
V = (6-2x)(8-2x)x
V = 4(3-x)(4-x)x
V = 4x(x^2-7x+12)
V = 4x^3-28x^2+48x
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b) graph the function and show the graph over the valid range of the variable x

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c) Using the graph, what is the value of x that will produce the maximum volume?
======================
Cheers,
Stan H.
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