Problema Solution
an opened- topped box can be created by cutting congruent square from each of the four corners of a piece of cardboard that has dimensions of 20cm by 30cm and folding up sides .Determine the dimension of squares that must be cut to create a box with a volume of 1008cm^3
Answer provided by our tutors
let 'x' be the side of the square that are being cut off then the dimensions of the box will b
length = 30-2x
width = 20-2x
20 - 2x > 0
x < 10
height = x
volume V = 1008 cm^3
V = length*width*height
x(30 - 2x)(20 - 2x) = 1008
multiply and simplify
x^3 - 25x^2 + 150x - 252 = 0
we can guess one of the roots by checking the factors of 252
252 = 2^2*3^2*7
since Vieta's formulas for cubic says that the product of the roots x1*x2*3 = - d/a and in our case it is - 252
and indeed for x = 3 we have
3^3 - 25*3^2 + 150*3 - 252 = 0
x1=3 is one root
if we divide (x^3 - 25x^2 + 150x - 252) by (x-3) we get (x^2 - 22x + 84) that is
(x - 3)(x^2 - 22x + 84) = 0
now we just need to find the roots of the quadratic
x^2 - 22x + 84 = 0
the roots are
x2 = 11 + 37^0.5
x3 = 11 - 37^0.5
we require x < 10 (otherwise, the width would be negative).
thus, 11 + √37 is discarded and 11 - √37 is valid.
therefore, the dimensions of the square can be 3 cm or 11 - √37 cm
you can either cut 3 cm squares or 11 - √37 cm squares off to yield the resulting volume.