A small mess of wasp has an exponential growth rate of 13% per month if the nest currently has 5,000 Watts the situation can be modeled by the equation w x t equals 5000 * 1.13t where w is the number of wasps after Time t is months which of the following statements is true about the equ

Question

I asked me increases w decreases quickly at first and then slowly
As t increases w increases slowly at first then quickly
St increases w increases quickly at first then slowly

Answer 1

To analyze the growth of the wasp population according to the given exponential growth model \( w(t) = 5000 \times 1.13^t \), we can consider how the population changes over time:

1. Exponential Growth Characteristics: In an exponential growth model, the growth rate is proportional to the current population. As time \( t \) increases, the population \( w(t) \) increases at an accelerating rate.

2. Initial Growth: At the very beginning, when \( t \) is small (e.g., \( t = 1, 2, 3 \)), the increase in population may seem modest, but as \( t \) continues to increase, the growth accelerates. For example:

   • At \( t=0 \): \( w(0) = 5000 \times 1.13^0 = 5000 \)
   • At \( t=1 \): \( w(1) = 5000 \times 1.13^1 = 5650 \)
   • At \( t=2 \): \( w(2) = 5000 \times 1.13^2 \approx 6403.50 \)
   • At \( t=3 \): \( w(3) = 5000 \times 1.13^3 \approx 7220.00 \)

3. Long-term Behavior: As \( t \) increases further, the increase in \( w(t) \) becomes significantly larger compared to earlier periods. This characteristic is typical of exponential growth.

Conclusion: The correct statement about the population growth is that as \( t \) increases, \( w \) increases slowly at first and then quickly (matching the understanding of exponential growth).

So the correct statement is:

"As \( t \) increases, \( w \) increases slowly at first and then quickly."