Extrema problem

S

sigma

Junior Member
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Feb 19, 2006
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106
Always have trouble with these types of questions.

A rectangular box with open top is to be constructed from a rectangular piece of cardboard 80 cm by 30 cm, by cutting out equal squares from each corner of the sheet of cardboard and folding up the resulting flaps. Find the dimensions of such a box of maximum volume.

Heres the picture:


Here's part of my work. I know how to do the question, the trouble is I get lost after I do the derivative because its a difficult function to factor. Can anybody show me how to do the factoring easily after my last line so that I can get the critical points?

\(\displaystyle \
\L\
\begin{array}{l}
V = x(30 - 2x)(80 - 2x) \\
V = (30x - 2x^2 )(80 - 2x) \\
\frac{{dV}}{{dx}} = (30 - 4x)(80 - 2x) + (30x - 2x^2 )( - 2) \\
0 = (30 - 4x)(80 - 2x) + (30x - 2x^2 )( - 2) \\
\end{array}
\\)
 
Why don't you simplfy your life and expand it, rather than try to carry the factors?

V(x) = 2400*x - 220*x^2 +4*x^3
V'(x) = 2400 - 440*x + 12*x^2

I am confident that you can now solve for the desired value.

Note: Back in algebra, when you were learning to factor, I suspect you managed to get the impression that factoring is the only way to go. Forget it. Use whatever form or process is most convenient. The answer doesn't care how you find it.
 
to make life easy let's use 3 decimeters by 8 decimeters ...

V = x(3 - 2x)(8 - 2x)

multiply it all out ...

V = x(4x^2 - 22x + 24)

V = 4x^3 - 22x^2 + 24x

dV/dx = 12x^2 - 44x + 24

12x^2 - 44x + 24 = 0

4(3x^2 - 11x + 6) = 0

4(3x - 2)(x - 3) = 0

only one solution is possible for the max ... x = 2/3 decimeter = 20/3 cm
 
Hello, sigma!

Your set-up is excellent!
That's at least 99% of the work . . . good work!

\(\displaystyle (30\,-\,4x)(80\,-\,2x) \,+\,(30x\,-\,2x^2 )(-2) \;=\;0\)
Click to expand...
We can't possible factor it in this form . . . we have to multiply it out.

But first . . . we can do some factoring and simplifying:

\(\displaystyle \;\;2(15\,-\,2x)2(40\,-\,x)\,+\,2x(15\,-\,x)(-2)\;=\;0\)

\(\displaystyle \;\;4(15\,-\,2x)(40\,-\,x)\,-\,4x(15\,-\,x) \;= \;0\)

Divide by 4: \(\displaystyle \,(15\,-\,2x)(40\,-\,x)\,-\,x(15\,-\,x)\;=\;0\)

\(\displaystyle \;\;600\,-\,15x\,-\,80x\,+\,2x^2\,-\,15x\,+\,x^2\;=\;0\)

We have the quadratic: \(\displaystyle \,3x^2\,-\,110x\,+600\;=\;0\)

\(\displaystyle \;\;\)which factors: \(\displaystyle \,(3x\,-\,20)(x\,-\,30)\;=\;0\)

\(\displaystyle \;\;\)and has roots: \(\displaystyle \,x\:=\:\frac{20}{3},\;30\)


Since we can't cut out 30-inch squares from each corner,
\(\displaystyle \;\;\)the answer is: \(\displaystyle \,x\,=\,\frac{20}{3}\,=\,6\frac{2}{3}\) cm
 
Thanks for all the help. But here's the problem. We are not allowed calculators on the final exam, so either factoring by the quadratic formula, or the decomposition method are going to be virtually impossible to do by hand (I assumed that's what everybody used but with calculators?) If I try to factor\(\displaystyle \
\L\
3x^2 - 110x + 600
\\) or \(\displaystyle \
\L\
12x^2 - 440x + 2400
\\) by either method, it would be way to difficult to do by hand. How on earth are you suppose to figure out the quadratic \(\displaystyle \
\L\
\frac{{ - 440 \pm \sqrt {(440)^2 - 4(12)(2400)} }}{{2(12)}}
\\) by hand? Does anybody here know what the square root of 78,400 is off hand because I sure don't. If I were to do this via the decomposition method I would have to know what factors are multiplied to get 28800 and also add up to -440. This would take far too long on any exam, let alone to do it without a calculator and make no mistakes. Sorry for my cynical response here but its questions like these that are just too demanding to do with out a calculator, and that's why I was wondering how it could be done easily through some factoring trick that's easy to do without a calculator. My prof showed me once a trick with a question like this but I can't figure it out. I just hope they don't ask one that difficult on the final.
 
soroban said:
We have the quadratic: \(\displaystyle \,3x^2\,-\,110x\,+600\;=\;0\)

\(\displaystyle \;\;\)which factors: \(\displaystyle \,(3x\,-\,20)(x\,-\,30)\;=\;0\)

\(\displaystyle \;\;\)and has roots: \(\displaystyle \,x\:=\:\frac{20}{3},\;30\)
Click to expand...

Soroban, how did you figure that out? Was there a trick used without a calculator? That's what you would get if the decomposition method was used but its too hard to do by hand. I wouldn't know what 2 factors are multiplied to get 1800 as well as add up to -110 (not without taking a really long time).
 
sigma ... you didn't learn a thing from my previous post, did you?
 
Sorry. Just wasn't obvious but that does work out really well and the decomposition method is easy to use now. I would have never thought of that though, and I guess that would only work when their perfect decimeters. (whole 10's). I'm going to take this problem to my prof and see what he thinks (see if he suggests the same method). But I still stand by what I said before. Unless this is the only way to make this factoring simpler, is there any other way?
 
You could try completing the square:

\(\displaystyle \L\\3(x-\frac{55}{3})^{2}-\frac{1225}{3}=0\)
 
sigma said:
\(\displaystyle \
\L\
12x^2 - 440x + 2400
\\)
Click to expand...
The only REAL answer is to encourage you to use any tools at your disposal.

You need experience at factoring - particularly at identifying Rational Roots. Quick passes at Synthetic Division, staying aware of upper bounds and Domain issues can whack the list of possibilities down very quickly.

Sometimes, just a quick look around is what it takes.

\(\displaystyle 440^{2}-4*12*2400\,=\,4^{2}*10^{2}*11^{2}-4*(4*3)*24*100\,=\,4^{2}*10^{2}*(11^{2}-3*24)\,=\,4^{2}*10^{2}*7^{2}\)

It's substantially easier to compute the square root of the last expression than the first.
 
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