Question

According to the Bureau of the Census, the population of the United States in 1990 was approximately 249 million and in 2000, 281 million. Use this information to predict the population for 2012.

Answer 1

Answer:

The prediction for the population in 2012 will be of 324.9 million.

Step-by-step explanation:

Exponential equation for population growth:

Considering that the population growth varies directly with time, it's value after t years is given by:

$P(t) = P(0)e^{kt}$

In which P(0) is the initial value and k is the growth rate.

The population of the United States in 1990 was approximately 249 million

This means that $P(0) = 249$, so:

$P(t) = P(0)e^{kt}$

$P(t) = 249e^{kt}$

In 2000, 281 million.

2000 - 1990 = 10, so $P(10) = 281$. We use this to find k.

$P(t) = 249e^{kt}$

$281 = 249e^{10k}$

$e^{10k} = \frac{281}{249}$

$\ln{e^{10k}} = \ln{\frac{281}{249}}$

$10k = \ln{\frac{281}{249}}$

$k = \frac{\ln{\frac{281}{249}}}{10}$

$k = 0.0121$

So

$P(t) = 249e^{0.0121t}$

Use this information to predict the population for 2012.

2012 - 1990 = 22, so this is P(22).

$P(22) = 249e^{0.0121*22} = 324.9$

The prediction for the population in 2012 will be of 324.9 million.