Question

The dimensions of a rectangle box are consecutive integers. If the box has volume of 13,800 cubic centimeters, what are its dimensions?

Answer 1

Three consecutive integers are x, x+1, x+2.
The volume of a rectangular box is the product of its three dimensions. The volume is 138000 cm³.

$x(x+1)(x+2)=13800 \\ (x^2+x)(x+2)=13800 \\ x^3+2x^2+x^2+2x=13800 \\ x^3+3x^2+2x-13800=0 \\ x^3-23x^2+26x^2-598x+600x-13800=0 \\ x^2(x-23)+26x(x-23)+600(x-23)=0 \\ (x^2+26x+600)(x-23)=0 \\ x^2+26x+600=0 \ \lor \ x-23=0 \\ \\ 1. \\ x^2+26x+600=0 \\ \\ a=1 \\ b=26 \\ c=600 \\ \Delta=b^2-4ac=26^2-4 \times 1 \times 600=676-2400=-1724 \\ \hbox{the discriminant is less than 0 so there are no real solutions} \\ \\ 2. \\ x-23=0 \\ x=23 \\ \\ x+1=23+1 =24 \\ \\ x+2=23+2=25$

The dimensions of the box are 23 cm x 24 cm x 25 cm.