Question

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Answer 1

0.66 inches of material is needed to be cut off to make the volume maximum.

maximum and minimum points test

When the second derivative of a function is negative, the function has a maximum point and if the second derivative is positive, the function has a minimum point.

Analysis:

After cut and folded, length = 8-2x

Width = 3-2x

Thickness = x.

Volume of the folded shape = (8-2x)(3-2x)(x)

After expansion, V = 4$x^{3}$-$22x^{2}$ +24x

for turning point of the function, dv/dx = 0

dv/dx = 12$x^{2}$ -44x + 24

lowest term = 3$x^{2}$ - 11x + 6

3$x^{2}$ - 11x + 6 = 0

3$x^{2}$ - 9x -2x +6 = 0

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

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

x = 2/3 or x = 3

To test for maximum point, we differentiate dv/dx again

we have 6x - 11

for x = 3,  6(3) - 11 = 18 - 11 = 7 which is positive x= 3 is a minimum

for x = 2/3 6(2/3) - 11 = 4 - 11 = -7, x = 2/3 is a maximum.

Therefore for maximum volume, the length to be cut out is 2/3 which is 0.66 inches.

Learn more about maximum and minimum points: brainly.com/question/26913652

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