Question

A box is formed by cutting congruent squares from the corners. Find the maximum volume of a box made out of a 14 inch by 18 inch cardboard. Find

the lenth of the side of the cut out square. Round to the nearest hundredth place "hint: you may have to change your window in desmos.
Jinches
a. Maximum volume
b. length of side of square: x=
inches

Answer 1

Explanation:

V = x(14 - 2x)(18 - 2x)

V = (14x - 2x²)(18 - 2x)

V = (252x - 28x² - 36x² + 4x³)

V = 4x³ - 64x² + 252x

V' = 12x² - 128x + 252

0 = 3x² - 32x + 63

x = (32 ±√(32² - 4(3)(63))) / (2(3)

x = (32 ± √268) / 6

x = 8.06 inches, which we ignore as this would mean we have no material left at all in the 14 inch dimension.

or

x = 2.60488... 2.60 inches

Vmax = 0.60(14 - 2(2.60))(18 - 2(2.60)) = 292.86478... = 292.86 inches³