Math Analysis Problem

Question

A candy box is made from a piece of cardboard that measures 17 by 10 inches. Squares of equal size will be cut out of each corner. The sides will then be folded up to form a rectangular box. What size square should be cut from each corner to obtain maximum​ volume?

Answer 1

Rectangular cardboard: length 17

width 10

Box: length = 17 - 2x

width = 10 - 2x

height = x

Volume = length * width * height = (17 - 2x)(10 - 2x)x

= (170 - 34x - 20x + 4x^2)x

= (170 - 54x + 4x^2)x

= 170x - 54x^2 + 4x^3

0= dV/dx = 170 - 108x + 12x^2

= 85 - 54x + 6x^2

= 6x^2 - 54x + 85

x = [54 +or- sqrt( 54^2 - 4*6*85) ]/ 12

= [ 54 +or- sqrt ( 876)]/12

= [ 54 +or- 2*sqrt(219)]/12

= [ 27 +or- sqrt(219)]/6

[(27 + sqrt(219)]/6 = 6.966441431158124

results in negative measures

[(27 - sqrt(219)]/6 = 2.033558568841876323819276161769

876 = 2 * 438 = 2 * 2 * 219 = 2 * 2 * 3 * 73