Question

A village experienced 2% population growth, compounded continuously, each year for 10 years. At the end of the 10 years, the population was 158.

1. What was the population of the village at the beginning of the 10 years according to the exponential growth function? Round your answer up to the next whole number, and do not include units.

Answer 1

The population of the village at the beginning of the 10 years was approximately 130. This is determined by using the formula for exponential growth that is continuously compounded and solving for the initial population.

Step-by-step explanation:

We can use the formula for exponential growth that is continuously compounded, which is P=Pe^rt, where P is the final population, P is the initial population, r is the growth rate, and t is the time.

In this case, we know that P=158, r=0.02 (2% expressed as a decimal), and t=10. We need to find P.

Substituting the given values into the formula, we get 158=Pe^(0.02*10).

We can calculate e^(0.02*10) using a calculator to get approximately 1.22. Therefore, the equation becomes 158=P*1.22.

To find P, we can divide both sides of the equation by 1.22 resulting in P ≈ 158/1.22 ≈ 129.5. Since we are asked to round up to the next whole number, the original population was approximately 130.

Learn more about Exponential Growth

Answer 2

The initial population at the beginning of the 10 years was 129.

Step-by-step explanation:

The population of the village may be modeled by the following function.

`P(t) = P_(0)e^(rt)`

In which P is the population after t hours,
`P_(0)` is the initial population and r is the growth rate, in decimal.

In this problem, we have that:

`P(10) = 158, r = 0.02`.

So

`158 = P_(0)e^(0.02*10)`

`P_(0) = 158*e^(-0.2)`

`P_(0) = 129`

The initial population at the beginning of the 10 years was 129.