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 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.

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.'

GPT-4o mini · Answer

To complete the input-output table based on the equation \( y = 4060(0.95)^t \), we will calculate the number of birds for each point in time (t) given:

For \( t = 0 \): \[ y = 4060(0.95)^0 = 4060(1) = 4060 \]
For \( t = 5 \): \[ y = 4060(0.95)^5 = 4060(0.77378) \approx 3135.78 \approx 3136 \]
For \( t = 10 \): \[ y = 4060(0.95)^{10} = 4060(0.59874) \approx 2421.86 \approx 2422 \]
For \( t = 15 \): \[ y = 4060(0.95)^{15} = 4060(0.46329) \approx 1886.79 \approx 1887 \]
For \( t = 20 \): \[ y = 4060(0.95)^{20} = 4060(0.34868) \approx 1411.72 \approx 1412 \]

Now we have the completed input-output table:

\[ \begin{array}{|c|c|} \hline \text{Years (t)} & \text{Number of Birds (y)} \ \hline 0 & 4060 \ 5 & 3136 \ 10 & 2422 \ 15 & 1887 \ 20 & 1412 \ \hline \end{array} \]

Next, we need to determine which graph matches this sequence of decay. The values of birds are decreasing, starting from 4060 down to approximately 1412 over a span of 20 years, indicating an exponential decay.

Based on the descriptions provided, we look for a graph that:

Shows a decreasing trend.
Begins around 4000 and decreases to around the low end (possibly below 1500) over time.

The appropriate choice would likely be the third graph, where the y-axis ranges from 40,000 to 52,000, accommodating the peak number of birds (4060). However, this would not seem to match since 4060 is significantly lower than this range.

By process of elimination, if the second graph accurately reflects a more appropriate range for bird counts (0 to 6000), it would show the correct decreasing pattern, starting from around 4000 and trending down.

Thus, the best match for the decay behavior described and the table calculated is the second graph with a y-axis ranging from 0 to 600 and displaying the correct decay trend.