Question

the table shows the amount of pet food in cups remaining in an automatic feeder as a function of the number of meals the feeder has dispensed.
number of meals dispensed, n: 1,3,6,7
amount of pet food remaining, f(n)(cups): 21,15,6,3
based on the table, which function models this situation?
f(x)= ______

Answer 1

To find a function \( f(n) \) that models the given data points, we observe the number of meals dispensed \( n \) and the corresponding amount of pet food \( f(n) \).

The data is as follows:
- \( n = 1 \), \( f(1) = 21 \)
- \( n = 3 \), \( f(3) = 15 \)
- \( n = 6 \), \( f(6) = 6 \)
- \( n = 7 \), \( f(7) = 3 \)

We notice that as the number of meals increases, the amount of pet food decreases. Let's look for a pattern.

1. Calculate the differences:
- From \( n = 1 \) to \( n = 3 \): \( 21 - 15 = 6\)
- From \( n = 3 \) to \( n = 6 \): \( 15 - 6 = 9\)
- From \( n = 6 \) to \( n = 7 \): \( 6 - 3 = 3\)

The differences between the amounts indicate a pattern but do not form an arithmetic sequence.

2. To determine a model, let’s assume a linear relationship and check if it fits.

Let’s use linear interpolation:
Setting up the points \((1, 21)\), \((3, 15)\), \((6, 6)\), and \((7, 3)\) appears to be decreasing linearly.

We can derive a linear equation using two points, for example, \( (1, 21) \) and \( (3, 15) \):
- The slope \( m \) can be calculated as:
\[
m = \frac{f(3) - f(1)}{3 - 1} = \frac{15 - 21}{3 - 1} = \frac{-6}{2} = -3
\]
- Now substituting into point-slope form \( f(n) = m(n - n_1) + f(n_1) \):
\[
f(n) - 21 = -3(n - 1)
\]
Expanding gives:
\[
f(n) = -3n + 3 + 21 = -3n + 24
\]

Let's validate if this equation holds for other points.
- For \( n = 3 \):
\[
f(3) = -3(3) + 24 = -9 + 24 = 15 \quad \text{(correct)}
\]
- For \( n = 6 \):
\[
f(6) = -3(6) + 24 = -18 + 24 = 6 \quad \text{(correct)}
\]
- For \( n = 7 \):
\[
f(7) = -3(7) + 24 = -21 + 24 = 3 \quad \text{(correct)}
\]

Thus, the function that models this situation is:

\[
f(n) = -3n + 24
\]

So, the final answer is:
\[
f(x) = -3x + 24
\]