Question

The population of a certain country in 1996 was 283 million people.

In addition, the population of the country was growing at a rate of 1.0 % per year. Assuming that this growth rate continues, the model P(t) = 283(1.010 )^t-1996 represents the population P (in millions of people) in year t. According to this model, when will the population of the country reach
A. 321 million? B. 366 million?

Answer 1

The population of the country will reach 321 million in approximately the year 2025, and it will reach 366 million in approximately the year 2027.

Step-by-step explanation:

To find when the population of the country will reach a certain number, we can substitute the given population value and growth rate into the model equation P(t) = 283(1.010 )^(t-1996). Let's solve for t when the population is 321 million:

321 = 283(1.010)^(t-1996)

Divide both sides by 283:

1.134 = 1.010^(t - 1996)

Take the natural logarithm of both sides:

ln(1.134) = ln(1.010^(t - 1996))

Use the property of logarithms to bring down the exponent:

ln(1.134) = (t - 1996) ln(1.010)

Divide both sides by ln(1.010):

t - 1996 = ln(1.134) / ln(1.010)

Add 1996 to both sides:

t = 1996 + ln(1.134) / ln(1.010)

Using a calculator, we find that t ≈ 2024.7, so the population will reach 321 million in approximately the year 2025.

Now let's solve for t when the population is 366 million:

366 = 283(1.010)^(t-1996)

Divide both sides by 283:

1.293 = 1.010^(t - 1996)

Take the natural logarithm of both sides:

ln(1.293) = ln(1.010^(t - 1996))

Use the property of logarithms to bring down the exponent:

ln(1.293) = (t - 1996) ln(1.010)

Divide both sides by ln(1.010):

t - 1996 = ln(1.293) / ln(1.010)

Add 1996 to both sides:

t = 1996 + ln(1.293) / ln(1.010)

Using a calculator, we find that t ≈ 2026.8, so the population will reach 366 million in approximately the year 2027.