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
.

(1 point)
Responses

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 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 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 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.

Answer 1

To determine which graph matches the decay rate of the bird population modeled by the equation \( y = 4,060(0.95)^t \), we need to calculate the values of \( y \) for the given years \( t \) (0, 5, 10, 15, 20).

Let's calculate the number of birds for each year:

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

2. For \( t = 5 \): \[ y = 4,060(0.95)^5 \approx 4,060(0.7738) \approx 3,138.51 \approx 3,139 \]

3. For \( t = 10 \): \[ y = 4,060(0.95)^{10} \approx 4,060(0.5987) \approx 2,426.80 \approx 2,427 \]

4. For \( t = 15 \): \[ y = 4,060(0.95)^{15} \approx 4,060(0.4633) \approx 1,883.45 \approx 1,883 \]

5. For \( t = 20 \): \[ y = 4,060(0.95)^{20} \approx 4,060(0.3487) \approx 1,415.42 \approx 1,415 \]

So the completed input-output table is approximately:

| Years (t) | Number of Birds (y) | |-----------|----------------------| | 0 | 4,060 | | 5 | 3,139 | | 10 | 2,427 | | 15 | 1,883 | | 20 | 1,415 |

Now, let’s consider the graphs described in the options. We are looking for a graph that shows a decreasing trend, consistent with a decay model, and has values that would align with a starting point around 4,060 and decreases to a lower number over time.

The key points to check:

• The first value is around 4,060.
• The subsequent values should reflect a decreasing pattern and align more closely with what we calculated.

Given these details, the correct graph would be the one displaying a curve that starts at approximately 4,060 and declines, representing the decay in bird population over the years.

Among the responses, the graph that shows the y-axis ranging broadly enough to fit values starting at 4,060 and dropping down to lower values (such as the range up to 6,000) is appropriate.

Based on the provided options, the most fitting response is likely:

• 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 6000. There is an arrow at the decreasing end.

This graph would effectively show the decay of the bird population as modeled by the given exponential decay equation.