Use the table to answer the question.

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
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(1 point)
Responses

A coordinate plane's x-axis ranges from 0 to 30 by 5-unit increments and its y-axis ranges from 0 to 6000 by 1000-unit increments. The x-axis is labeled 'Years' and the y-axis is labeled 'Number of Birds.'
Image with alt text: A coordinate plane's x-axis ranges from 0 to 30 by 5-unit increments and its y-axis ranges from 0 to 6000 by 1000-unit increments. The x-axis is labeled 'Years' and the y-axis is labeled 'Number of Birds.'

A curve with a decreasing pattern is plotted on a graph with 'Years' as the x-axis ranging from 0 to 10 and 'Number of Birds' on the y-axis ranging from 0 to 6000. There is an arrow at the decreasing end.
Image with alt text: A curve with a decreasing pattern is plotted on a graph with 'Years' as the x-axis ranging from 0 to 10 and 'Number of Birds' on the y-axis ranging from 0 to 6000. There is an arrow at the decreasing end.

A curve with a decreasing pattern is plotted on a graph with 'Years' as the x-axis ranging from 0 to 30 in increments of 1 and 'Number of Birds' on the y-axis ranging from 0 to 600 in increments of 20. There is an arrow at the decreasing end.
Image with alt text: A curve with a decreasing pattern is plotted on a graph with 'Years' as the x-axis ranging from 0 to 30 in increments of 1 and 'Number of Birds' on the y-axis ranging from 0 to 600 in increments of 20. There is an arrow at the decreasing end.

A curve with a decreasing pattern is plotted on a graph with 'Years' as the x-axis ranging from 0 to 30 in increments of 1 and 'Number of Birds' on the y-axis ranging from 40,000 to 52,000 in increments of 400. There is an arrow at the decreasing end.
Image with alt text: A curve with a decreasing pattern is plotted on a graph with 'Years' as the x-axis ranging from 0 to 30 in increments of 1 and 'Number of Birds' on the y-axis ranging from 40,000 to 52,000 in increments of 400. There is an arrow at the decreasing end.
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option 1 2 or 3 4

Answer 1

To determine which graph matches the species of bird modeled by the equation \( y = 4,060(0.95)^t \), we first need to compute the number of birds for the given years (t).

Using the equation:

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

2. For \( t = 5 \): \[ y = 4,060(0.95)^5 \approx 4,060 \times 0.77378 \approx 3,138.77 \text{ (rounded to 3,139)} \]

3. For \( t = 10 \): \[ y = 4,060(0.95)^{10} \approx 4,060 \times 0.59874 \approx 2,426.00 \text{ (rounded to 2,426)} \]

4. For \( t = 15 \): \[ y = 4,060(0.95)^{15} \approx 4,060 \times 0.46329 \approx 1,884.84 \text{ (rounded to 1,885)} \]

5. For \( t = 20 \): \[ y = 4,060(0.95)^{20} \approx 4,060 \times 0.34868 \approx 1,414.37 \text{ (rounded to 1,414)} \]

Thus, we have the following coordinates for the input-output table:

• (0, 4,060)
• (5, 3,139)
• (10, 2,426)
• (15, 1,885)
• (20, 1,414)

Now, we can determine which graph matches the sequence of the decay rate. We look for a graph that displays a decreasing pattern starting from around 4,060 and decreasing to values below 2,000 as t increases to 20.

Given this examination, it appears the graphic responses will reflect the same properties:

1. The first graph ranges up to 6000 but does not show a detailed view of lower values.
2. The second graph only captures up to 6000 and is limited to only 10 years, which is not sufficient.
3. The third graph, ranging from 0 to 600 (not sufficient to reflect the values), appears to misrepresent the numbers.
4. The fourth graph has a decreasing pattern ranging from around 40,000 to 52,000 and, again, is not suitable to represent our values.

None of the options listed perfectly represent the bird populations over the years given from the equation. However, we can identify that the characteristic of exponential decay should be prevalent in any adequate selection. If these options still crystallize only partial aspects of the expected results, based on your data closely approximations and decay values in lower ranges can lead toward the reasonable solution.

To wrap it up, look for a graph that captures a significant downward trend starting around 4000 and reflects decreasing values through 0 to 30 years - meaning the overriding graphical detail is critical for identification under exponential tendencies.