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A certain population of beetles is increasing at a rate of 22% per month. If there are currently 350 beetles, the equation f(x)=350(1.22)x
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can be used to model the situation. Finish the chart and plot the points to determine which graph correctly displays the situation.(1 point)
Responses

Image with alt text: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 350 in increments of 50.

Image with alt text: An increasing curve with an arrow on the top is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. The y-axis for Beetles ranges from 0 to 2000 in increments of 250.

Image with alt text: A decreasing curve with an arrow at the end is plotted on a coordinate plane. The x-axis for months ranges from 0 to 10 in increments of 1. Th

1 answer

To complete the chart and find the correct graph for the beetle population growth modeled by the equation \( f(x) = 350(1.22)^x \), we need to calculate the population for \( x = 0, 1, 2, 3, 4, 5 \).

  1. For \( x = 0 \): \[ f(0) = 350(1.22)^0 = 350 \times 1 = 350 \]

  2. For \( x = 1 \): \[ f(1) = 350(1.22)^1 = 350 \times 1.22 = 427 \]

  3. For \( x = 2 \): \[ f(2) = 350(1.22)^2 \approx 350 \times 1.4884 \approx 520.94 \approx 521 \]

  4. For \( x = 3 \): \[ f(3) = 350(1.22)^3 \approx 350 \times 1.8047 \approx 631.645 \approx 632 \]

  5. For \( x = 4 \): \[ f(4) = 350(1.22)^4 \approx 350 \times 2.1336 \approx 746.76 \approx 747 \]

  6. For \( x = 5 \): \[ f(5) = 350(1.22)^5 \approx 350 \times 2.6014 \approx 910.49 \approx 910 \]

Now we can summarize the values in the chart:

| x (Months) | f(x) (Beetles) | |------------|----------------| | 0 | 350 | | 1 | 427 | | 2 | 521 | | 3 | 632 | | 4 | 747 | | 5 | 910 |

With these values, we can observe that the population of beetles is increasing exponentially.

In terms of the graphs:

  • The correct graph should be an increasing curve (since the population is increasing) and not a decreasing curve.
  • The y-axis should have values greater than 350, extending upwards to around 1000 or more to accommodate the values calculated above, particularly for \( f(5) \).

Thus, the right image would be the one that shows an increasing curve with an arrow extending upwards, corresponding with the growth in beetle population over time.