The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 5% in 10 years. What will be the population in 20 years

551 people

Step-by-step explanation:

We have that for this type of situation the equation that models the population depending on the time is:

P (t) = I * e ^ (k * t)

Where I is the initial population, that is to say that for this case it would be:

P (t) = 500 * e ^ (k * t)

k is the constant of proportionality and t the time, they tell us that in 10 years it increases 5%, therefore the population would be:

500 + 500 * 0.05 = 525, therefore:

525 = 500 * e ^ (k * 10)

we solve:

525/500 = e ^ (k * 10)

ln 1.05 = k * 10

k = 0.00488

Now, to calculate how much the population is in 20 years it would be:

P (t) = 500 * e ^ (0.00488 * 20)

P (t) = 551.26

In other words, the population would be approximately 551 people