SOLUTION: POLYNOMIAL BOX Let's say that I have a 50cm x 60cm cardboard. My problem/task is to design a box (open top) that will get the greatest capacity or maximum volume of the cardboar

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Question 800985: POLYNOMIAL BOX
Let's say that I have a 50cm x 60cm cardboard. My problem/task is to design a box (open top) that will get the greatest capacity or maximum volume of the cardboard can get. Thus, the size of the square to be cut will be my main problem such that the box will have a maximum capacity or volume. Each side of the square is represented by x. So my question is, what is the value of x so that the volume of the box will reach its maximum?
OPEN TOP
Height = (x)
Length = 60-2x
Width = 50-2x
Since, V=Length x Width x Height
P(x)= (60-2x)(50-2x)(x)
Thanks in Advance! :).
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Let's say that I have a 50cm x 60cm cardboard. My problem/task is to design a box (open top) that will get the greatest capacity or maximum volume of the cardboard can get. Thus, the size of the square to be cut will be my main problem such that the box will have a maximum capacity or volume. Each side of the square is represented by x. So my question is, what is the value of x so that the volume of the box will reach its maximum?
OPEN TOP
Height = (x)
Length = 60-2x
Width = 50-2x
Since, V=Length x Width x Height
P(x)= (60-2x)(50-2x)(x)
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P(x) = 4x^3 - 220x^2 + 3000x
Find the max of the function.
P'(x) = 12x^2 - 440x + 3000 = 0
3x%5E2+-+110x+%2B+750+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B-110x%2B750+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-110%29%5E2-4%2A3%2A750=3100.

Discriminant d=3100 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--110%2B-sqrt%28+3100+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-110%29%2Bsqrt%28+3100+%29%29%2F2%5C3+=+27.6129406047167
x%5B2%5D+=+%28-%28-110%29-sqrt%28+3100+%29%29%2F2%5C3+=+9.05372606194996

Quadratic expression 3x%5E2%2B-110x%2B750 can be factored:
3x%5E2%2B-110x%2B750+=+%28x-27.6129406047167%29%2A%28x-9.05372606194996%29
Again, the answer is: 27.6129406047167, 9.05372606194996. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-110%2Ax%2B750+%29

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x = ~ 9.04 gives a volume of ~12096 cc
The 27... value is a local minimum.
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The function is not factorable, so I don't see a way to do it without using the derivative.