Chapter 9.9, Problem 57PFA

a.

To determine

To find a function to represent number of population to after x years.

Answer to Problem 57PFA

  $f(x)=82400{\left(1+\frac{2.5}{100}\right)}^x$

Explanation of Solution

Given information:

In 2010, O has population $82400$

Rate of increase is $2.5\%$

Formula used:

people after t years

  $=P{\left(1+\frac{r}{100}\right)}^t$

P is initial people

r rate of increase of people

t is time in years

Calculation:

Initial people $=82400$

Rate of increase of people per month $=2.5\%$

Using formula function of sport blog after t years

  $f(x)=82400{\left(1+\frac{2.5}{100}\right)}^x$

b.

To determine

To find a function to represent number of popuplation to the after x years.

Answer to Problem 57PFA

  $g(x)=75600{\left(1-\frac{4.7}{100}\right)}^x$

Explanation of Solution

Given information:

In 2010, E has population 75600

Decrease rate is 300 resident per mont

Formula used:

people after t years

  $=P{\left(1-\frac{r}{100}\right)}^t$

P is initial people

r rate of increase of people

t is time in years

Calculation:

Initial people $=75600$

Rate of increase of people per month $=300$

Rate of increase of people 12 month $=3600$

Using formula

  $75600-3600=75600{\left(1-\frac{r}{100}\right)}^1$

On solving,

  $720=756\left(1-\frac{r}{100}\right)$

On solving,

  $\frac{r}{100}=\left(1-\frac{720}{756}\right)$

On solving,

  $r=\left(\frac{3600}{756}\right)$

On solving,

  $r=4.7\%$

Function

  $g(x)=75600{\left(1-\frac{4.7}{100}\right)}^x$

c.

To determine

To find $(f-g)(x)=f(t)-g(t)$ .

Answer to Problem 57PFA

  $(f-g)(x)=82400{\left(1+\frac{2.5}{100}\right)}^x{}^{}-75600{\left(1-\frac{4.7}{100}\right)}^x$

Explanation of Solution

Given information:

function of O

  $f(x)=82400{\left(1+\frac{2.5}{100}\right)}^x$

function of E

  $g(x)=75600{\left(1-\frac{4.7}{100}\right)}^x$

Calculation:

function of O

  $f(x)=82400{\left(1+\frac{2.5}{100}\right)}^x$

function of E

  $g(x)=75600{\left(1-\frac{4.7}{100}\right)}^x$

  $(f-g)(x)=f(t)-g(t)$

Substituting value of function of O and function of E

  $(f-g)(x)=82400{\left(1+\frac{2.5}{100}\right)}^x{}^{}-75600{\left(1-\frac{4.7}{100}\right)}^x$

It represent difference of O and E

d.

To determine

To find the value of $(f-g)(6)$ .

Answer to Problem 57PFA

  $(f-g)(6)=38924$

Explanation of Solution

Given information:

Find the value of $(f-g)(6)$

  $(f-g)(x)=82400{\left(1+\frac{2.5}{100}\right)}^x{}^{}-75600{\left(1-\frac{4.7}{100}\right)}^x$

Calculation:

  $(f-g)(x)=82400{\left(1+\frac{2.5}{100}\right)}^x{}^{}-75600{\left(1-\frac{4.7}{100}\right)}^x$

On solving,

  $(f-g)(6)=82400{\left(1+\frac{2.5}{100}\right)}^6-75600{\left(1-\frac{4.7}{100}\right)}^6$

On solving,

  $(f-g)(6)=82400{\left(1.025\right)}^6-75600{\left(95.5\right)}^6$

On solving,

  $(f-g)(6)=38924$

Population difference between O and E after 6 years