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 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.

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.'
Skip to navigation

Answer 1

To answer the question, we first need to calculate the values of the number of birds (y) for the given years (t) using the provided equation:

\[ y = 4,060(0.95)^t \]

Let's calculate for each year:

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

2. At t = 5: \[ y = 4,060(0.95)^5 \approx 4,060(0.7748) \approx 3,143.21 \] Rounding to the nearest whole number: \( y \approx 3,143 \)

3. At t = 10: \[ y = 4,060(0.95)^{10} \approx 4,060(0.5987) \approx 2,427.38 \] Rounding to the nearest whole number: \( y \approx 2,427 \)

4. At t = 15: \[ y = 4,060(0.95)^{15} \approx 4,060(0.4633) \approx 1,880.66 \] Rounding to the nearest whole number: \( y \approx 1,881 \)

5. At t = 20: \[ y = 4,060(0.95)^{20} \approx 4,060(0.3487) \approx 1,415.24 \] Rounding to the nearest whole number: \( y \approx 1,415 \)

Now, the completed input-output table is:

| Years (t) | Number of Birds (y) | |-----------|----------------------| | 0 | 4,060 | | 5 | 3,143 | | 10 | 2,427 | | 15 | 1,881 | | 20 | 1,415 |

Analysis of the Graphs

Since the number of birds decreases over time, we are looking for a graph with a steadily declining curve that starts at around 4,060 and decreases to around 1,415 by 20 years.

1. The first option shows a decreasing pattern, but covers a x-axis range up to 30 years. We must check if it starts near 4,060 and decreases correctly.
2. The second option also only shows up to 10 years. It might not show the entire trend as it does not reach 20 years.
3. The third option also extends from 40,000 to 52,000, which is not relevant to our bird population.
4. The fourth option has the correct axes (from 0 to 30 for x and up to 6000 for y which encompasses our values from years 0 to 20).

Given the values we calculated and the axes range, the best option would be:

The first option with the x-axis ranging from 0 to 30 (in increments of 1) and y-axis from 0 to 600 (in increments of 20).

This option captures the trend across the entire specified range and has a decreasing pattern as required.