To calculate how long it will take for the town's population to reach 20700, we can use the formula for exponential growth:
\( P = P_0 \times (1 + r)^t \)
Where:
- P is the future population we want to reach (20700)
- \( P_0 \) is the initial population (15000)
- r is the growth rate (4.5% or 0.045 as a decimal)
- t is the time in years we want to find
Substitute these values into the formula:
\( 20700 = 15000 \times (1 + 0.045)^t \)
Solve for t:
\( \frac{20700}{15000} = (1.045)^t \)
\( 1.38 = (1.045)^t \)
To solve for t, we can take the natural logarithm (ln) of both sides:
\( ln(1.38) = ln((1.045)^t) \)
Using the property of logarithms, we can bring down the exponent:
\( ln(1.38) = t \times ln(1.045) \)
Now, divide by ln(1.045) to solve for t:
\( t = \frac{ln(1.38)}{ln(1.045)} \)
Calculate this to find the approximate time in years it will take for the population to reach 20700.
