Question

The exponential model A = 438 e⁰.⁰¹⁵ ᵗ describes the​ population, a, of a country in​ millions, t years after 2003. use the model to determine when the population of the country will be 557 million.

Answer 1

To determine when the population will be 557 million, the exponential model A = 438e^(0.015t) is used. By setting A to 557 and solving for t, we find that t is approximately 18.57, meaning the population will reach 557 million around the year 2022.

Step-by-step explanation:

Solving for Time in an Exponential Population Model

To solve the mathematical problem completely and determine when the population of a country will be 557 million, we use the given exponential model for population growth, A = 438e0.015t. In this case, A represents the population in millions, and t is the number of years after 2003. Plugging in the population value of 557 million, we aim to solve for t.

First, we write down our equation with A set to 557:

557 = 438e0.015t

Next, we divide both sides by 438:

557 / 438 = e0.015t

Now we calculate the left side of the equation:

1.2717 ≈ e0.015t

To solve for t, we take the natural logarithm of both sides:

ln(1.2717) = 0.015t

Finally, we divide by 0.015 to isolate t:

t ≈ ln(1.2717) / 0.015

Using a calculator to compute the natural logarithm and the division:

t ≈ 18.57

Therefore, the population of the country is expected to reach 557 million approximately 18.57 years after the year 2003, which is around the year 2022.