SOLUTION: An open-top box is to be constructed from a 4 foot by 6 foot rectangular cardboard by cutting out equal squares at each corner and folding up the flaps. Let x denote the length of
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Question 86483: An open-top box is to be constructed from a 4 foot by 6 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.
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.
You can put this solution on YOUR website! An open-top box is to be constructed from a 4 foot by 6 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.
a) Find the function V that represents the volume of the box in terms of x.
Answer:
Drawing a picture would be extremely helpful.
1. Draw a 4 by 6 foot rectangle
2. Make boxes at each of the four corners with sides x feet
Length: 6 - 2x
Width: 4 - 2x
Height: x
V = product of all sides
V = (6 - 2x)(4 - 2x)(x)
V = 4(3 - x)(2 - x)(x)
V = 4(6 - 5x + x^2)(x)
V = 4(6x - 5x^2 + x^3)
V = 24x - 20x^2 + 4x^3
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.
Approx. when x = 1
~~~~
V(x) = 24x - 20x^2 + 4x^3
V'(x) = 24 - 40x + 12x^2
0 = 6 - 10x + 3x^2
~
Exact:
You can put this solution on YOUR website! An open-top box is to be constructed from a 4 foot by 6 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.
:
We know from the above information:
Box length = (6-2x)
Box width = (4-2x)
Box height = x
:
a) Find the function V that represents the volume of the box in terms of x.
Answer:
Vol = length * width * height
V = (6-2x)*(4-2x)*x
V = x(24 - 20x + 4x^2); FOILed (6-2x)(4-2x); then mult by x
V = 4x^3 - 20x^2 + 24x; represents the volume of the box
:
b) Graph this function and show the graph over the valid range of the variable x..
Plot the value from .2 to 1.8 only; y = V
x | y
-------
.2 | 4.032
.4 | 6.656
.6 | 8.064
.8 | 8.448
1.0| 8.0
1.2| 6.912
etc
:
Show Graph here.
c) Using the graph, what is the value of x that will produce the maximum volume?
Answer.
:
It looks like max volume will occur when x = .8
:
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