Asked by Mishaka

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)
Answered by Mishaka
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?
Answered by bobpursley
set V'=0, and you have a quadratic. Why not use the quadratic equation..

12x^2+88x-112=0
3x^2+22x-28=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.
Answered by Mishaka
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?
Answered by bobpursley
I don't see how you got the signs as you did. Please recheck
Answered by bobpursley
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.
Answered by bobpursley
Ok, your equation is right. Recheck your final signs as I stated.
Answered by Mishaka
I rechecked and found that 3x^2-22x+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).
Answered by bobpursley
Let me do some thinking...
if 0=V' = 112 - 88x - 12x^2
multipy both sides by -1, and
12x^2+88x-112=0

I don't see those as your signs....
Answered by Mishaka
Now I'm lost, I don't get why you changed the signs.
Answered by bobpursley
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

Answered by Mishaka
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.
Answered by Mishaka
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!
Answered by Noah M
Mishaka is correct, the exact root is (11-√37)/3
There are no AI answers yet. The ability to request AI answers is coming soon!