Chapter 5.6, Problem 97AYU

A Population Model The resident population of the United States in 2015 was 320 million people and was growing at a rate of $0.7\%$ per year. Assuming that this growth rate continues, the model $P\left(t\right)=320{\left(1.007\right)}^{t-2015}$ represents the population $P$ (in millions of people) in year $t$.

(a) According to this model, when will the population of the United States be 400 million people?

(b) According to this model, when will the population of the United States be 435 million people?

Source: U.S Census Bureau

To determine

To find:

a. According to this model, when will the population of the United States be 400 million people?

Answer to Problem 97AYU

a. 2047

Explanation of Solution

Given:

The resident population of the United States in 2015 was 320 million people and was growing at a rate of $0.7\%$ per year. Assuming that this growth rate continues, the model $P\left(t\right)=320{\left(1.007\right)}^{t-2015}$ represents the population $\text{P}$ (in millions of people) in year $t$.

Calculation:

$P\left(t\right)=320{\left(1.007\right)}^{t-2015}$

a. Determine when will the population of the United States be 400 million people.

$400=320{\left(1.007\right)}^{t-2015}$

$\frac{400}{320}={\left(1.007\right)}^{t-2015}$

$1.25={\left(1.007\right)}^{t-2015}$

$\log 1.25=\log {\left(1.007\right)}^{t-2015}$

$\log 1.25=\left(t-2015\right)\log 1.007$

$\frac{\log 1.25}{\log 1.007}=t-2015$

$31.9891=t-2015$

$t=2015+32$

$t=2047$

To determine

To find:

b. According to this model, when will the population of the United States be 435 million people?

Answer to Problem 97AYU

b. 2059

Explanation of Solution

Given:

The resident population of the United States in 2015 was 320 million people and was growing at a rate of $0.7\%$ per year. Assuming that this growth rate continues, the model $P\left(t\right)=320{\left(1.007\right)}^{t-2015}$ represents the population $\text{P}$ (in millions of people) in year $t$.

Calculation:

$P\left(t\right)=320{\left(1.007\right)}^{t-2015}$

b. Determine when will the population of the United States be 435 million people.

$435=320{\left(1.007\right)}^{t-2015}$

$\frac{435}{320}={\left(1.007\right)}^{t-2015}$

$1.3594={\left(1.007\right)}^{t-2015}$

$\log 1.3594=\log {\left(1.007\right)}^{t-2015}$

$\log 1.3594=\left(t-2015\right)\log 1.007$

$\frac{\log 1.3594}{\log 1.007}=t-2015$

$44.01=t-2015$

$t=2015+44$

$t=2059$