Exponential Functions - Basic (Solving) Question A town has a population of 12000 and grows at 2% every year. To the nearest year, how long will it be until the population will reach 16200? square

To solve this problem, we need to find the number of years it takes for the population to grow from 12,000 to 16,200, given that the population grows at 2% every year.<br /><br />Given information:<br />- Initial population: 12,000<br />- Target population: 16,200<br />- Annual growth rate: 2% (0.02)<br /><br />We can use the exponential growth formula to calculate the number of years:<br /><br />\[ P(t) = P_0 \times (1 + r)^t \]<br /><br />Where:<br />- \( P(t) \) is the population at time \( t \)<br />- \( P_0 \) is the initial population<br />- \( r \) is the annual growth rate<br />- \( t \) is the number of years<br /><br />Substituting the given values, we get:<br /><br />\[ 16,200 = 12,000 \times (1 + 0.02)^t \]<br /><br />Simplifying the equation:<br /><br />\[ 1.35 = (1.02)^t \]<br /><br />To solve for \( t \), we can take the natural logarithm (ln) of both sides:<br /><br />\[ \ln(1.35) = \ln((1.02)^t) \]<br /><br />Using the property of logarithms that \( \ln(a^b) = b \ln(a) \), we get:<br /><br />\[ \ln(1.35) = t \ln(1.02) \]<br /><br />Solving for \( t \):<br /><br />\[ t = \frac{\ln(1.35)}{\ln(1.02)} \]<br /><br />Calculating the values:<br /><br />\[ \ln(1.35) \approx 0.3001 \]<br />\[ \ln(1.02) \approx 0.0198 \]<br /><br />\[ t \approx \frac{0.3001}{0.0198} \approx 15.15 \]<br /><br />Rounding to the nearest year, the population will reach 16,200 in approximately 15 years.<br /><br />Therefore, it will take approximately 15 years for the population to reach 16,200.