Question

In Exercises \(21-32,\) use an exponential model to solve the problem. Population Growth The population of Knoxville is \(500,000\) and is increasing at the rate of 3.75\(\%\) each year. Approximately when will the population reach 1 million?

Short answer

After performing the calculation in the final step, you find that it will take approximately 18.6 years for the population to reach 1 million.

Step-by-step solution

Convert the growth rate to decimal form

The first thing to do is convert the growth rate from a percentage to a decimal. This is done by dividing the percentage by 100. Using this method, the growth rate of 3.75% becomes 0.0375.

Setup the exponential growth formula

Next, we will substitute the values into the exponential growth formula. We have: \( P_0 = 500000 \), the initial population; \( r = 0.0375 \); and \( P(t) = 1000000 \), the future population that we want to reach. Plugging these values in, we get \( 1000000 = 500000(1 + 0.0375)^t \).

Simplification and logarithmic transformation

To make the equation easier to solve, we first simplify the right-hand side by dividing both sides by 500000, which results in \( 2 = (1.0375)^t \). Then we apply the logarithm base \( e \), or ln, on both sides. The equation becomes \( ln (2) = t * ln (1.0375) \). In other words, \( t = ln (2) / ln (1.0375) \).

Solving for t

To solve for \( t \), all you need to do now is to divide the natural logarithm of 2 by the natural logarithm of 1.0375.