Open Box calculating 4 squares (sides)

Question

An open box is constructed from a 28 cm by 21 cm sheet of cardboard by cutting out equal squares of side x cm from each corner and then folding up the sides. The volume of such a box is given by the cubic function V(x) = x(28-2x)(21-2x). Using technology, determine 

a. What size of square, correct to the nearest hundredth, is cut from the corners to have a box 

with a volume of 750 cm? State both possible solutions. 

b. What size of square, correct to the nearest hundredth, is cut from the corners to have a box 

with the maximum volume?

Answer 1

750 = x(28-2x)(21-2x) = 4x^3 -98x^2 + 588x

4x^3 -98x^2 +588x -750 = 0

2x^3 -49x^2 +294x -375 = 0

by decartes rule of signs, there are max 3 positive real solutions

and no real negative solutions

graph it using desmos graphing calculator and look where the graph intersects the x axis

at least one solution is between 1 and 2

closer to 2 than to 1

2(8)-49(4)+294(2) -375

=16-196 +588-375

= 604 -571

= 33

2 -49+294-375

= 296-424

= -128

33 is much closer to 0 than -128 is, so x is much closer to 2 than to 1

for maximum volume

4x^3 -98x^2 +58x

f'(x) = 12x^2 -196x +58 = 0

divide by 2:

6x^2 -98x +29 = 0

use the quadratic formula: x=-b/2a +/-(1/2a)sqr(b^2-4ac), a=6, b=-98, c=29

x = 98/12 +/-(1/12)sqr(196^2 -4(12)(58))

= (-98+/-188.79)/12

= about 90.79/12

= about 7.57 inches