SOLUTION: A rectangular piece of cardboard is 13 inches long and 9 inches wide. From each corner a square piece is cut out, and then the flaps are turned up to form an open box. Determine th

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Question 370483: A rectangular piece of cardboard is 13 inches long and 9 inches wide. From each corner a square piece is cut out, and then the flaps are turned up to form an open box. Determine the length of a side of the square pieces so that the volume of the box is as large as possible.
i got this:
V=(13-2h)(9-2h)(h)
but what do i do next?
Found 3 solutions by stanbon, Edwin McCravy, jsmallt9:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
A rectangular piece of cardboard is 13 inches long and 9 inches wide. From each corner a square piece is cut out, and then the flaps are turned up to form an open box. Determine the length of a side of the square pieces so that the volume of the box is as large as possible.
i got this:
V=(13-2h)(9-2h)(h)
but what do i do next?
---
Find the derivative:
dV/dh = ?
---
Then solve dV/dh = 0
----
Use the 2nd derivative test to determine at what "h" you have a maximum.
-----
Cheers,
Stan H.
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Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
You then 

1. Multiply out the right side

2. Find dV%2Fdh 

3. Set dV%2Fdh = 0.

4. Show that it is a maximum. 

--------------------------------------

1.
V=%2813-2h%29%289-2h%29%28h%29

V=%28117-26h-18h%2B4h%5E2%29%28h%29

V=%28117-44h%2B4h%5E2%29%28h%29

V=117h-44h%5E2%2B4h%5E3

2.
dV%2Fdh=117-88h%2B12h%5E2

3.
117-88h%2B12h%5E2=0

12h%5E2-88h%2B117=0

h+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+



h+=+%2888+%2B-+sqrt%287744-5616%29%29%2F24+

h+=+%2888+%2B-+sqrt%282128%29%29%2F24+

h+=+%2888+%2B-+sqrt%2816%2A133%29%29%2F24+

h+=+%2888+%2B-+4sqrt%28133%29%29%2F24+

h+=+%284%2822+%2B-+sqrt%28133%29%29%29%2F24+


h+=+%2822+%2B-+sqrt%28133%29%29%2F6+

using the + we get h = 5.588760432
using the - we get h = 1.744572901

4.
Using the 2nd derivative test.

dV%2Fdh=117-88h%2B12h%5E2
d%5E2V%2Fdh%5E2=-88%2B24h

Substituting h = 5.588760432 gives a positive second derivative
which means the graph is concave upward there and is a minimum. But
we want a maximum.  [We could also tell that this could not be the
answer because we would be cutting more than half of the shorter side
and this would be impossible.)

Substituting h = 1.744572901 gives a negative second derivative
which means the graph is concave downward there and is a maximum.

Therefore the correct answer is  h = %2822+%2B-+sqrt%28133%29%29%2F6
or h = 1.744572901. 

This will gives a box of approximately 91.43817871 cubic inches.

Edwin

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
V=(13-2h)(9-2h)(h)
Great work so far. To find the h that makes a box of maximum volume we will be finding the derivative of your volume function. (If you are not taking Calculus or you have never heard of a derivative, then you might as well stop reading and repost your question. If you do repost you might want to specify "without using Calculus" so other tutors don't give you Calculus-based solutions as well.)

In general, maximum and/or minimum values for a function are found at any of the following:
  • At the endpoints of the domain. For your volume function h must be between 0 and 4.5. We can't have negative sides to the box. If h < 0, then the height is negative. If h >4.5, then the 9-2h side will be negative. If h=0 then the volume is zero. And if h=4.5 then the 9-2h side is zero making the volume zero again. So the endpoints of your domain tell you minimum values for the volume.
  • Where the first derivative is either zero or undefined. So we need to find the first derivative:
    The derivative of your volume function will be a little easier if we go ahead and multiply it out first:
    V+=+%28117-44h%2B4h%5E2%29%28h%29
    V+=+117h-44h%5E2%2B4h%5E3
    Now we can find the first derivative:
    V' = 117-88h%2B12h%5E2
    Now we will use the derivative to find relative maximum and/or minimum points. These will occur where the first derivative is zero or undefined. This derivative is defined for all possible values of h. But we can find where it is zero:
    0+=+117-88h%2B12h%5E2
    This will not factor so we will need to use the Quadratic Formula:
    h+=+%28-%28-88%29+%2B-+sqrt%28%28-88%29%5E2+-4%2812%29%28117%29%29%29%2F2%2812%29
    which simplifies as follows:
    h+=+%28-%28-88%29+%2B-+sqrt%287744+-4%2812%29%28117%29%29%29%2F2%2812%29
    h+=+%28-%28-88%29+%2B-+sqrt%287744+-+5616%29%29%2F2%2812%29
    h+=+%28-%28-88%29+%2B-+sqrt%282128%29%29%2F2%2812%29
    h+=+%2888+%2B-+sqrt%282128%29%29%2F24
    h+=+%2888+%2B-+sqrt%2816%2A133%29%29%2F24
    h+=+%2888+%2B-+sqrt%2816%29%2Asqrt%28133%29%29%2F24
    h+=+%2888+%2B-+4%2Asqrt%28133%29%29%2F24
    h+=+%284%2822+%2B-+sqrt%28133%29%29%29%2F%284%2A6%29
    h+=+%28cross%284%29%2822+%2B-+sqrt%28133%29%29%29%2F%28cross%284%29%2A6%29
    h+=+%2822+%2B-+sqrt%28133%29%29%2F6
    In long form this is:
    h+=+%2822+%2B+sqrt%28133%29%29%2F6 or h+=+%2822+-+sqrt%28133%29%29%2F6
    If you get out your calculator you will find that the first h is greater than 4.5. This is not in the domain so we reject it. The other value for h, %2822+-+sqrt%28133%29%29%2F6 then is the one which must create the box of maximum volume. This is true because
    • The endpoints of the domain create minimum volume boxes.
    • The first derivative is defined for all values of h so no maximum or minimum values come from an undefined first derivative.
    • Therefore, the only h in the domain that makes the first derivative zero, %2822+-+sqrt%28133%29%29%2F6, is our answer because a maximum must exist at one of these three places.