Asked by anonymous
A sheet of cardboard 25 cm by 40 cm will be made into an open-topped box by cutting equal-sized squares from each corner and folding up the four edges. what will be the dimensions of the box with the largest volume?
Answers
Answered by
Reiny
The classic question that just about every Calculus text uses as an example of
problems dealing with optimization.
Let the side of the cut-out-square be x cm
So dimensions of the finished box are 25-2x by 40-2x for the base and x for the height
V = x(25-2x)(40-2x)
= 4x^3 - 130x^2 + 100x
dV/dx = 12x^2 - 260x + 1000 = 0 for a max of V
3x^2 - 65x + 250 = 0
x = ...... use whatever method is your favourite to solve quadratics
remember 0 < x < 12.5 for the dimensions to make sense
problems dealing with optimization.
Let the side of the cut-out-square be x cm
So dimensions of the finished box are 25-2x by 40-2x for the base and x for the height
V = x(25-2x)(40-2x)
= 4x^3 - 130x^2 + 100x
dV/dx = 12x^2 - 260x + 1000 = 0 for a max of V
3x^2 - 65x + 250 = 0
x = ...... use whatever method is your favourite to solve quadratics
remember 0 < x < 12.5 for the dimensions to make sense
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