Step-by-step explanation:
To solve this, we can use the formula for exponential growth:
\[ P(t) = P_0 \times (1 + r)^t \]
Where:
- \( P(t) \) is the final population after time \( t \),
- \( P_0 \) is the initial population (450 in this case),
- \( r \) is the growth rate per time period (2% or 0.02 in this case),
- \( t \) is the time period in weeks.
We want to find \( t \) when \( P(t) = 1060 \). Let's plug in the values and solve for \( t \):
\[ 1060 = 450 \times (1 + 0.02)^t \]
\[ \frac{1060}{450} = (1.02)^t \]
\[ 2.3556 \approx (1.02)^t \]
Now, we need to solve for \( t \). Taking the natural logarithm of both sides:
\[ \ln(2.3556) \approx t \times \ln(1.02) \]
\[ t \approx \frac{\ln(2.3556)}{\ln(1.02)} \]
\[ t \approx \frac{0.859}{0.0198} \]
\[ t \approx 43.44 \]
So, it would take approximately 43.44 weeks for the number of rabbits to increase from 450 to 1060 without pest control. We can round this to 44 weeks.
