A paper company needs to ship paper to a large printing business. The paper will be shipped in small boxes and large boxes. The volume of each small box is 6 cubic feet and the volume of each large box is 12 cubic feet. There were twice as many small boxes shipped as large boxes shipped and the total volume of all boxes was 168 cubic feet. Write a system of equations that could be used to determine the number of small boxes shipped and the number of large boxes shipped. Define the variables that you use to write the system.

A paper company needs to ship paper to a large printing business. The paper will be shipped in small boxes and large boxes. The volume of each small box is 6 cubic feet and the volume of each large box is 12 cubic feet. There were twice as many small boxes shipped as large boxes shipped and the total volume of all boxes was 168 cubic feet. Write a system of equations that could be used to determine the number of small boxes shipped and the number of large boxes shipped. Define the variables that you use to write the system.

\underline{\text{Define Variables:}}

Define Variables:

May choose any letters.

\text{Let }s=

Let s=

\,\,\text{the number of small boxes shipped}

the number of small boxes shipped

\text{Let }l=

Let l=

\,\,\text{the number of large boxes shipped}

the number of large boxes shipped

Each small box has a volume of 6 cubic feet, so the volume of ss small boxes would be 6s6s cubic feet. Each large box has a volume of 12 cubic feet, so the volume of ll large boxes would be 12l12l cubic feet. Therefore, the total volume 6s+12l6s+12l equals 168:168:

6s+12l=168

6s+12l=168

Since there were twice as many small boxes shipped as large boxes, there were more small boxes, so if we multiply 2 by the number of large boxes, we will get the number of small boxes, meaning ss equals 2l.2l.

let s = number of small boxes shipped

let l= number of large boxes shipped

6s+121 = 168

s=21

table 3 represents a direct variation function. : )

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for