The population, P, of the city of Hazelton has grown according to the mathematical model P=32000(1.09)^t, where t is the number of years since 1990. a) What do the numbers 32000 and 1.09 represent? b) If this trend continues, in what year will the population reach 100000? Justify your answer.

## Step 1: Interpreting the constants in the model<br />### The given model is $P = 32000(1.09)^t$. The number 32000 represents the initial population in the year 1990 (when $t=0$). The number 1.09 represents the growth factor, indicating a 9% annual growth rate (1.09 = 1 + 0.09).<br /><br />## Step 2: Setting up the equation to find the year when the population reaches 100000<br />### We want to find $t$ when $P = 100000$. So, we set up the equation $100000 = 32000(1.09)^t$.<br /><br />## Step 3: Solving for $t$<br />### Divide both sides by 32000: $\frac{100000}{32000} = (1.09)^t$, which simplifies to $3.125 = (1.09)^t$. Taking the logarithm of both sides (base 10): $\log(3.125) = t \log(1.09)$. Solving for $t$: $t = \frac{\log(3.125)}{\log(1.09)} \approx \frac{0.49485}{0.03743} \approx 13.22$.<br /><br />## Step 4: Determining the year<br />### Since $t$ represents the number of years since 1990, we add 13.22 to 1990. Since we are looking for a full year, we round up to the next whole number, which is 14. Therefore, the population will reach 100000 in the year $1990 + 14 = 2004$.