SOLUTION: Please Help!!! I cant figure this out. 1) 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 the

Question 67627: Please Help!!! I cant figure this out.
1) 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 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.

Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
1) 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 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:
The dimensions of the box will be (4-2x) by (6-2x) by x
:
FOIL (4-2x)(6-2x) and you have: 24 - 20x + 4x^2, mult that x,
:
V(x) = 4x^3 - 20x^2 + 24x
:
:
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.
It looks like that about x =.75 ft will produce max volume (8.4 cu ft)
:
You can find out exactly for x = .75: (4-1.5) * (6-1.5) * .75 = 8.4375
:
My Ti83 gave the max at x = .78475 produces a max vol of 8.45 cu ft
:
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