To complete the table based on the formula provided, \( y = 4,060(0.95)^t \), we need to calculate the number of birds for each value of \( t \) (0, 5, 10, 15, and 20).
Let’s compute the values step-by-step.
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For \( t = 0 \): \[ y = 4,060(0.95)^0 = 4,060 \times 1 = 4,060 \]
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For \( t = 5 \): \[ y = 4,060(0.95)^5 = 4,060 \times 0.77378 \approx 3,145.68 \text{ (round to 3,146)} \]
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For \( t = 10 \): \[ y = 4,060(0.95)^{10} = 4,060 \times 0.59874 \approx 2,427.61 \text{ (round to 2,428)} \]
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For \( t = 15 \): \[ y = 4,060(0.95)^{15} = 4,060 \times 0.46329 \approx 1,883.06 \text{ (round to 1,883)} \]
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For \( t = 20 \): \[ y = 4,060(0.95)^{20} = 4,060 \times 0.34868 \approx 1,415.86 \text{ (round to 1,416)} \]
Now we can summarize our findings in the completed table:
\[ \begin{array}{c|c} \text{Years} (t) & \text{Number of Birds} (y) \ \hline 0 & 4,060 \ 5 & 3,146 \ 10 & 2,428 \ 15 & 1,883 \ 20 & 1,416 \ \end{array} \]
Now that we have our calculated values, we can match them to one of the graph descriptions provided.
Looking at the nature of \( y = 4,060(0.95)^t \), we see that it models an exponential decay. The number of birds decreases over time, starting high and lowering gradually.
Among the graph descriptions:
- The first option describes a graph up to 6000, which does not match as our highest value is 4060.
- The second option describes a large scale up to 52,000, which also does not match.
- The third option, with a y-axis ranging from 0 to 600 and incremented in 20s, suggests lower values which fit our calculated results.
- The fourth option also suggests a smaller y-axis up to 6000, but over a limited year span (0 to 10).
The best match is the third option which describes a curve with decreasing values on a y-axis from 0 to 600 in increments of 20, indicating a decay consistent with our results.
Therefore, the correct answer is the third option that describes the graph with a curve showing decreasing numbers of birds.
4 answers