Posted by Mishaka on Friday, December 16, 2011 at 8:20pm.
A rectangular piece of cardboard, 8 inches by 14 inches, is used to make an open top box by cutting out a small square from each corner and bending up the sides. What size square should be cut from each corner for the box to have the maximum volume?
So far I have: V = (14  2x)(8  2x)(h)

Calculus (Optimization, Still Need Help)  Mishaka, Friday, December 16, 2011 at 8:43pm
I just wanted to correct something for my equation, it should be: V = (14  2x)(8  3x)(x), which simplifies to V = 112x  44x^2  4x^3.
Take the derivative:
V' = 112  88x  12x^2
Now all I need are the roots, any help? I think I found one around 1.10594, possibly? 
Calculus (Optimization)  bobpursley, Friday, December 16, 2011 at 8:48pm
set V'=0, and you have a quadratic. Why not use the quadratic equation..
12x^2+88x112=0
3x^2+22x28=0
x= (22+sqrt (22^2+12*28))/6
doing it in my head, I get about..
(22+28)/6= 4/6, 7.5 in my head.
check my work and estimates. 
Calculus (Optimization)  Mishaka, Friday, December 16, 2011 at 9:00pm
I think that you might have gotten the equation wrong, I think that it should be: 3x^2  22x + 28. When I put this equation into the quadratic equation, I got 5.694254177 and 1.639079157. So the squares that need to be cut out should have an area of approximately 2.69 square inches?

Calculus (Optimization)  bobpursley, Friday, December 16, 2011 at 9:04pm
I don't see how you got the signs as you did. Please recheck

Calculus (Optimization)  bobpursley, Friday, December 16, 2011 at 9:06pm
hang on, I reread the problem statement. In the first response I gave, I took your equation. I don't think it is right.
give me a minute. 
Calculus (Optimization)  bobpursley, Friday, December 16, 2011 at 9:07pm
Ok, your equation is right. Recheck your final signs as I stated.

Calculus (Optimization)  Mishaka, Friday, December 16, 2011 at 9:12pm
I rechecked and found that 3x^222x+28 has the correct signs. Knowing this equation and the values I found from the quadratic equation, would you say that the 1.639079157 term is correct? (The 2.69 square inches came from squaring the 1.639079157).

Calculus (Optimization)  bobpursley, Friday, December 16, 2011 at 9:17pm
Let me do some thinking...
if 0=V' = 112  88x  12x^2
multipy both sides by 1, and
12x^2+88x112=0
I don't see those as your signs.... 
Calculus (Optimization)  Mishaka, Friday, December 16, 2011 at 9:19pm
Now I'm lost, I don't get why you changed the signs.

Calculus (Optimization)  bobpursley, Friday, December 16, 2011 at 9:22pm
You are lost. This is algebra.
if 0=112  88x  12x^2
do whatever you know to put it in standard form, ax^2+bx+c=0
when you do that a and b will have the SAME signs. Surely you can do that.
if a=12, then b=88, and c=112
if a=12, then b=88, and c=112 
Calculus (Optimization)  Mishaka, Friday, December 16, 2011 at 9:28pm
Okay, so does this change my original answer of approximately 1.64 to 4.42??? The 4.42 came from putting your new values in the quadratic equation.

Calculus (Optimization)  Mishaka, Friday, December 16, 2011 at 9:31pm
Nevermind, that 4.42 was a mistake and my very original answer of 1.105940354 was absolutely correct!!! This is the right answer, I know it!