To create the input-output table for the equation \( y = 4,060(0.95)^t \), we will calculate the number of birds \( y \) at each given time \( t \). We substitute the values of \( t \) (0, 5, 10, 15, and 20) into the equation.
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For \( t = 0 \): \[ y = 4,060(0.95)^0 = 4,060(1) = 4,060 \]
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For \( t = 5 \): \[ y = 4,060(0.95)^5 \approx 4,060(0.77378) \approx 3,139.01 \quad \text{(round to 3,139)} \]
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For \( t = 10 \): \[ y = 4,060(0.95)^{10} \approx 4,060(0.59874) \approx 2,426.05 \quad \text{(round to 2,426)} \]
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For \( t = 15 \): \[ y = 4,060(0.95)^{15} \approx 4,060(0.46329) \approx 1,882.50 \quad \text{(round to 1,883)} \]
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For \( t = 20 \): \[ y = 4,060(0.95)^{20} \approx 4,060(0.34868) \approx 1,415.15 \quad \text{(round to 1,415)} \]
Now, we can summarize the results in the input-output table:
| Years (t) | Number of Birds (y) | |-----------|----------------------| | 0 | 4,060 | | 5 | 3,139 | | 10 | 2,426 | | 15 | 1,883 | | 20 | 1,415 |
Analyzing the Decay Rate
The decay represented by the equation \( y = 4,060(0.95)^t \) indicates an exponential decay where each successive time period leads to a reduction in the number of birds by 5% (since \( 0.95 \) represents 95% of the previous number).
Graph Matching
To identify which graph matches the sequence, look for a graph that starts at 4,060 and steadily decreases, reflecting the calculated values above, specifically noticing a curve that drops sharply at first and then more slowly approaches the x-axis, consistent with exponential decay.
If you have the provided graphical choices, find the one that matches this pattern !
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