Volume - Maths Question (year 10 advanced class)

Question

Start with a rectangular sheet of paper, measuring 8 cm by 11 cm. Cut out squares from each corner. What is the length of the sides of each square that is cut out in order to create a box with maximum volume? Give your answer to the nearest millimetre.

Answer 1

paper 8 by 11 with squares cut from each corner to maximize volume

let x = the side of each square, then

volume = (8-2x)(11-2x)x = (88-38x +4x^2)x = 88x -38x^2 +4x^3

take the derivative and set = 0

V'(x) = 88 -76x +12x^2 = 0

3x^2 - 19x - 22 = 0

use the quadratic formula

x = 19/6 +/- (1/6)sqr(19^2 +12(22))

= (19 +/- sqr25)/6 = 24/6 or 14/6 = 4 or 2 1/3 cm

for x = 4, V=0 which is a minimum Volume

for x= 2 1/3 or 7/3 cm = 2.333.. cm = 23.33 mm = about 23 mm

V = (8-14/3)(11-14/3)(14/3)

= (10/3)(19/3)(14/3)

= 190(14)/27

= about 98.5 cm^3 = maximum volume possible