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:
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For \( t = 0 \): \[ y = 4,060(0.95)^0 = 4,060 \times 1 = 4,060 \]
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For \( t = 5 \): \[ y = 4,060(0.95)^5 \approx 4,060 \times 0.77378 \approx 3,139.88 \approx 3,140 \]
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For \( t = 10 \): \[ y = 4,060(0.95)^{10} \approx 4,060 \times 0.59874 \approx 2,430.78 \approx 2,431 \]
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For \( t = 15 \): \[ y = 4,060(0.95)^{15} \approx 4,060 \times 0.46329 \approx 1,884.17 \approx 1,884 \]
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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.
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