For constant speed or rate problems, use: total amount $=$ rate $\times$ time. Which part of the function should represent the rate, and which part the time?

Pick a simple value like $t=1$ minute. In 1 minute, how many pages should the printer produce? Which function gives that number when you plug in $t=1$?

In Desmos, enter each option as a separate expression, but replace $t$ with $1$ (for 1 minute): p(1)=1/12, p(1)=12/1, p(1)=12*1, and p(1)=1+12. Think about how many pages the printer should produce in 1 minute (using 12 pages per minute) and see which expression matches that value.

Now try $t=2$ minutes by entering each as p(2)=2/12, p(2)=12/2, p(2)=12*2, and p(2)=2+12. Decide which expression’s outputs correctly show that the printer adds the same number of pages (12) every additional minute.

The printer works at a constant rate of 12 pages per minute. This means:

• In 1 minute, it prints 12 pages.
• In 2 minutes, it prints $2\times 12=24$ pages.
• In $t$ minutes, it will print $t$ groups of 12 pages.

So the number of pages depends on both the rate (12 pages per minute) and the time in minutes ($t$).

For any constant rate problem, the basic relationship is:

• Total amount $=$ rate $\times$ time

Here,

• Total amount $=$ number of pages $p(t)$
• Rate $=$ 12 pages per minute
• Time $=$ $t$ minutes

So the expression for the number of pages after $t$ minutes should be $12$ times $t$.

We want the function where the output (pages) equals $12$ times the input ($t$ minutes). That function is

This is the model that correctly represents 12 pages per minute.