Chapter 1.3, Problem 22E

(a)

To determine

To find: The estimated population in $1915$ and $1940$ .

Answer to Problem 22E

The estimated population in $1915$ is 12,315 and $1940$ is 24,265.

Explanation of Solution

Given information: In 1890, the population of Silver Run was $6250$ and the population increased at a rate of $2.75\%$ per year.

Calculation:

The initial population of Silver Run was $6250$ in the year $1890$ . The population of Silver Run is increasing at a rate of $2.75\%$ per year. This means that the population becomes $1.0275$ times every year.

The exponential function for the population of Silver Run is:

  $\begin{array}{c}y=y_0{a}^t\\ y=6250{\left(1.0275\right)}^t\end{array}$

The initial value for $t$ is zero in the year $1890$ . The value of $t$ from year 1890 to 1915 is:

  $t=25$

The value of $t$ from year 1890 to 1940 is:

  $t=50$

Substitute $25$ for $t$ in the exponential model.

  $y=6250{\left(1.0275\right)}^{25}$

Use the calculator to find the value of $6250{\left(1.0275\right)}^{25}$ . Press “ON” button and enter the keystrokes as given below:

  $\boxed{6250}\boxed{\times }\boxed{(}\boxed{1.0275}\boxed{)}\boxed{^}\boxed{25}$

The required value is $12314.75$ . In 1915, the approximate population was 12315.

Substitute 50 for $t$ in the exponential model.

  $y=6250{\left(1.0275\right)}^{50}$

Use the calculator to find the value of $6250{\left(1.0275\right)}^{50}$ . Press “ON” button and enter the keystrokes as given below:

  $\boxed{6250}\boxed{\times }\boxed{(}\boxed{1.0275}\boxed{)}\boxed{^}\boxed{50}$

The required value is $24264.51$ . In 1940, the approximate population was 24,265.

Therefore, the estimated population in $1915$ was 12,315 and $1940$ was 24,265.

(b)

To determine

To find: The number of years taken to reach the population to 50,000.

Answer to Problem 22E

The number of years taken to reach the population to 50,000 is $77$ years.

Explanation of Solution

Given information: In 1890, the population of Silver Run was $6250$ and the population increased at a rate of $2.75\%$ per year.

Calculation:

As calculated in part (a), the exponential function for the population of Silver Run is $y=6250{\left(1.0275\right)}^t$ .

Substitute 50,000 for $y$ in the exponential function.

  $\begin{array}{c}y=6250{\left(1.0275\right)}^t\\ 50000=6250{\left(1.0275\right)}^t\end{array}$

Follow the steps to solve the equation by calculator:

• First press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key.
• Enter right hand side of the equation $6250{\left(1.0275\right)}^t$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{6250}\boxed{\times }\boxed{(}\boxed{1.0275}\boxed{)}\boxed{^}\boxed{\text{X}}$ . Now, press the key $\boxed{\text{ENTER}}$ .
• Enter right hand side of the equation $50000$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{50000}$ .
• Press $\boxed{\text{WINDOW}}$ key and adjust the window at $\left[-20,100\right]$ by $\left[-100,55000\right]$ .
• Now, press keystrokes given below to find the value of $t$ as:

  $\boxed{2\text{nd}}\to \boxed{\text{TRACE}}\to \boxed{5}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}$

  

So, the value of $\text{X}$ is $76.651$ years. The approximate value is $77$ .

Therefore, the number of years taken to reach the population to 50,000 is $77$ years.