Chapter 1.3, Problem 21E

To determine

To find: The number of years for the population to reach $1$ million.

Answer to Problem 21E

The population will reach $1$ million in about $19$ years.

Explanation of Solution

Given information: The population of Knoxville is $500,000$ and is increasing at the rate of $3.75\%$ each year.

Calculation:

The initial population of Knoxville is $500,000$ .

The population of Knoxville is increasing at the rate of $3.75\%$ each year. So, the population becomes $1.0375$ times every year.

Let, the required number of years be $t$ . The exponential function for the population of Knoxville is:

  $y=500000{\left(1.0375\right)}^t$

Substitute $1,000,000$ for $y$ in the exponential function.

  $\begin{array}{c}y=500000{\left(1.0375\right)}^t\\ 1000000=500000{\left(1.0375\right)}^t\end{array}$

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 $500000{\left(1.0375\right)}^t$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{500000}\boxed{\times }\boxed{(}\boxed{1.0375}\boxed{)}\boxed{^}\boxed{\text{X}}$ . Now, press the key $\boxed{\text{ENTER}}$ .
• Enter right left side of the equation $1000000$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{1000000}$ .
• Press $\boxed{\text{WINDOW}}$ key and adjust the window at $\left[-20,30\right]$ by $\left[-100,1100000\right]$ .
• 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 $18.828$ . The approximate value is 19.

Therefore, the population will reach $1$ million in about $19$ years.