To model the growth of the wasp nest with an exponential growth rate, we can use the formula:
\[ w(t) = w_0 \cdot e^{rt} \]
Where:
- \( w(t) \) is the population at time \( t \),
- \( w_0 \) is the initial population (5000 wasps),
- \( r \) is the growth rate (13% per month, or 0.13 as a decimal),
- \( t \) is the time in months,
- \( e \) is the base of the natural logarithm.
Substituting the values into the equation, we get:
\[ w(t) = 5000 \cdot e^{0.13t} \]
Analyzing the Growth:
For the statements about the growth behavior:
-
As t increases, w increases slowly at first and then quickly - This statement is true for exponential growth. In the beginning, the population increases slowly because the base (initial population) is small relative to its growth potential. However, as \( t \) increases and the population grows, the effect of the growth rate leads to increasingly larger increases.
-
As t increases, w increases quickly at first and then slowly - This statement does not accurately describe exponential growth.
-
As t increases, w decreases slowly at first and then quickly - This statement is not applicable because the population is growing, not decreasing.
-
As t increases, w decreases quickly at first and then slowly - This statement is also not applicable for the same reason as above.
Conclusion:
The correct statement about the wasp population is:
As t increases, w increases slowly at first and then quickly.
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