Margie is packing 108 small boxes into a large carton. the small boxes will fill all of the space inside the large carton. Each small box is 3 inches long 2 inches wide and 1 inch high. The width of the base, and the height of the large carton are the same. the length of the base is less then 36 inches. All of the dimensions are whole numbers. Explain how to find possible dimensions for the large carton.

2 inches wide

that means that the height of the large carton is 2 inches

find the volume of the carton

V=108 times volume of small box

V=108 times 1 times 2 time s3

V=648 in^2

height is 2

V=LWH

648=LW2 (note, this is not LW^2 but 2 times LW)

divie both sides by 2

324=LW

legnth of base is less than 36

div

we just need to find the values of legnth and width such that LW=324 and L<36 and that L or W is divisible by 3 (since each box is 3 inches long)

so lets factor 324

324=2*2*3*3*3*3

group them and find possible values for legnth and width less than 36 and divsible by 3

possible values

(L, W, H) possible values are

(27,12,2)

(18,18,2)

the possible dimentions are

27 by 12 by 2 or

18 by 18 by 2