SOLUTION: ______8______
|_|x
|x
|
13|
|
|_
|_|__________
This is a paper that is cut from EACH side in small squares when you fold it is a rec
Algebra.Com
Question 531437: ______8______
|_|x
|x
|
13|
|
|_
|_|__________
This is a paper that is cut from EACH side in small squares when you fold it is a rectangular prism with no top, 13 is the lenght and 8 is the width. it did'nt let me draw the other half but it is a FULL rectangle and all four sides of the corners a cube is cut out and that is X. What is the value of X and Maximum volume of it and could you show work too my Teacher said use a TI calculator if you can and i dont have one and she said that it was going to be a parabola it would be great if you could help me thanks!
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
This is a paper that is cut from EACH side in small squares when you fold it is a rectangular prism with no top, 13 is the lenght and 8 is the width. it did'nt let me draw the other half but it is a FULL rectangle and all four sides of the corners a cube is cut out and that is X. What is the value of X and Maximum volume of it and could you show work too my Teacher said use a TI calculator if you can and i dont have one and she said that it was going to be a parabola it would be great if you could help me thanks!
------
Draw a rectangle with length=13 and width=8
----
Draw a square at each corner of the rectangle (in the rectangle)
------------
Draw the rectangle that connects the squares
with width = 8-2x and length = 13-2x
----
If you cut out those squares and fold up the four sides
you will have a rectangular prism with height = x and
base = 8-2x by 13-2x
----
Volume = x(8-2x)(13-2x) = (4x^3 - 42x^2 + 104x)
-----
To find the maximum volume find the derivative:
V'(x) = 12x^2-84x+104
-----
Solve 12x^2-84x+104 = 0
3x^2 - 21x + 26 = 0
Solve using the Quadratic Formula:
x = [21 +- sqrt(21^2 - 4*3*26)]/6
---
x = [ 21 +- sqrt(129)]/2
---
Positive solution:
x = [21 + 11.36]/2
x = 16.18
---
Find the Maximum Volume
V(x) = (4x^3 - 42x^2 + 104x)
V(16.18) = 4*16.18^3 - 42*16.18^2 + 104*16.18 = 7630.6 cu. units
========================================
Cheers,
Stan H.
============
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Once the box is folded, the length will be 
, the width will be 
, and the height will be 
. Since the volume of a rectangular prism is given by length times width times height, we create the volume function:
\ =\ (13\ -\ 2x)(8\ -\ 2x)(x)\ =\ 4x^3\ -\ 42x^2\ +\ 104x)
The graph of which is NOT a parabola, although the portion of the graph of interest, that is where is within an interval that allows for an actual physical box to be created, sorta-kinda looks like a parabola.
In order to actually create a box, must be in the interval )
. If were negative you would be dealing with the absurdity of cutting a negative amount from the corners. If 
, then you aren't cutting anything away from the corners and all you have is a flat 13 by 8 piece of paper with zero volume. If 
then \ =\ 0)
and again, you have zero volume.
Somewhere in between is a maximum volume, and we know this because of the Mean Value Theorem.
If a function is continuous and twice differentiable over an interval and if the first derivitive of the function is equal to zero at some point in the interval, then there is a local extremum at that point. If the second derivitive is negative at that point, then the extremum is a maximum.
Set the derivitive equal to zero:
Solve this quadratic and select the value that is within the previously discussed interval. This is the value that gives the extreme volume. Evaluate the original volume function for this value to get the actual extreme volume.
Take the second derivitive:
and evaluate it at the value of that gives the extreme volume. The second derivitive must be negative if the extreme point is a maximum.
John
My calculator said it, I believe it, that settles it
RELATED QUESTIONS
if you take square of cardboard with sides that are 8 feet long, and cut congruent... (answered by josgarithmetic)
Given a 10 cm by 20 cm piece of paper, if you cut out four equal squares from the corners (answered by Theo)
A small paper box is to be made from a sheet of paper measuring 20cm by 12 cm. The sides... (answered by josgarithmetic)
assume we take a sheet of paper that is 8.5 by 11 inches. we will transform it into a... (answered by ankor@dixie-net.com)
If you have a peice of paper that is 84" x 84". You are trying to make a box our of it... (answered by solver91311)
Equipment
- MathResources Graphing Tool (provided)
- 8 pieces of 8.5 x 11 paper
-... (answered by Alan3354)
You have a rectangular piece of wood that is 12 in by 6 in. A small length, x,is cut from (answered by ikleyn,josgarithmetic)
An open-top box is to be constructed from a 6 by 8 foot rectangular cardboard by cutting... (answered by venugopalramana)
An open-top box is to be constructed from a 6 by 8 foot rectangular cardboard by cutting... (answered by mukhopadhyay)
