Question

Colin is painting figurines. He spends 20 minutes painting each figurine. After painting for 60 minutes, he still has 9 more figurines left to paint.
The number
\[f\] of figurines left to paint is a function of
\[t\], the amount of time in minutes Colin spends painting.
Write the function's formula.
\[f=\]

Answer 1

To solve for the function \( f \) that represents the number of figurines Colin has left to paint as a function of time \( t \) (in minutes), we can break down the problem as follows:

1. Colin spends 20 minutes painting each figurine.
2. After painting for 60 minutes, he has 9 figurines left.

First, let's determine how many figurines Colin has painted after 60 minutes. Since each figurine takes 20 minutes to paint, we can calculate the number of painted figurines:

\[
\text{Number of figurines painted} = \frac{\text{Total time spent painting}}{\text{Time per figurine}} = \frac{60 \text{ minutes}}{20 \text{ minutes/figurine}} = 3 \text{ figurines}
\]

Let \( N \) be the total initial number of figurines Colin had. After painting 3 figurines, he has 9 left, which means:

\[
N - 3 = 9 \implies N = 12
\]

Thus, Colin started with 12 figurines.

Now, if we let \( t \) be the total time in minutes that Colin spends painting, the number of figurines he is able to paint is:

\[
\text{Number of figurines painted} = \frac{t}{20}
\]

Therefore, the number of figurines left to paint after \( t \) minutes is:

\[
f(t) = N - \frac{t}{20}
\]

Substituting \( N = 12 \) gives us:

\[
f(t) = 12 - \frac{t}{20}
\]

To express this in a more standard form, you can also write:

\[
f(t) = 12 - 0.05t
\]

Thus, the function for the number of figurines left to paint as a function of time \( t \) is:

\[
\boxed{f(t) = 12 - \frac{t}{20}}
\]