Question 800985
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^2 - 110x + 750 = 0}}}
*[invoke solve_quadratic_equation 3,-110,750]
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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.