Question

Use the table to answer the question.

Years (t) 0 5 10 15 20
Number of Birds (y)
Complete the input-output table and determine which graph matches the sequence of the decay rate of a specific species of bird modeled by the equation y=4,060(0.95)t .

Answer 1

To complete the input-output table based on the equation \( y = 4,060(0.95)^t \), we need to calculate the number of birds (\( y \)) at each value of \( t \) in the table.

We will calculate \( y \) for the following values of \( t \):

- \( t = 0 \)
- \( t = 5 \)
- \( t = 10 \)
- \( t = 15 \)
- \( t = 20 \)

Let's calculate each:

1. **For \( t = 0 \)**:
\[
y = 4,060(0.95)^0 = 4,060 \times 1 = 4,060
\]

2. **For \( t = 5 \)**:
\[
y = 4,060(0.95)^5 \approx 4,060 \times 0.77378 \approx 3,139.88 \approx 3,140
\]

3. **For \( t = 10 \)**:
\[
y = 4,060(0.95)^{10} \approx 4,060 \times 0.59874 \approx 2,430.78 \approx 2,431
\]

4. **For \( t = 15 \)**:
\[
y = 4,060(0.95)^{15} \approx 4,060 \times 0.46329 \approx 1,884.17 \approx 1,884
\]

5. **For \( t = 20 \)**:
\[
y = 4,060(0.95)^{20} \approx 4,060 \times 0.34868 \approx 1,415.29 \approx 1,415
\]

Now, we can complete the input-output table:

\[
\begin{array}{|c|c|}
\hline
\text{Years (t)} & \text{Number of Birds (y)} \\
\hline
0 & 4,060 \\
5 & 3,140 \\
10 & 2,431 \\
15 & 1,884 \\
20 & 1,415 \\
\hline
\end{array}
\]

### Summary of the Data

- At \( t = 0 \), \( y = 4,060 \)
- At \( t = 5 \), \( y = 3,140 \)
- At \( t = 10 \), \( y = 2,431 \)
- At \( t = 15 \), \( y = 1,884 \)
- At \( t = 20 \), \( y = 1,415 \)

### Graphing Interpretation

The function \( y = 4,060(0.95)^t \) represents an exponential decay, as the base (0.95) is less than 1, indicating that the number of birds decreases over time. The graph of this function would show a downward curve, getting closer to zero as \( t \) increases, but never actually reaching zero.

If you have options for the graph, you would select the one that shows an exponential decay pattern similar to what is detailed above.