Chapter 1.3, Problem 32E

(a)

To determine

To find: The number of years taken to reduce the number of cases to $1000$ .

Answer to Problem 32E

The number of years taken to reduce the number of cases to $1000$ is $10.319$ years.

Explanation of Solution

Given information: The number of cases are reduced by $20\%$ every year and the number of cases today are $10000$ .

Calculation:

The number of cases is reducing by $20\%$ each year. So, the number of cases reduced to $80\%$ of the total case each year that is the number of cases becomes $0.8$ times every year.

The present number of cases are $10000$ .

Substitute $10000$ for $y_0$ and $0.8$ for $a$ in the exponential function for the number of cases left.

  $\begin{array}{l}y=y_0{a}^t\\ y=10000{\left(0.8\right)}^t\end{array}$

Substitute $y=1000$ in the exponential function.

  $\begin{array}{c}y=10000{\left(0.8\right)}^t\\ 1000=10000{\left(0.8\right)}^t\\ 1=10{\left(0.8\right)}^t\end{array}$

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 $10{\left(0.8\right)}^t$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{10}\boxed{\times }\boxed{(}\boxed{0.8}\boxed{)}\boxed{^}\boxed{\text{X}}$ . Now, press the key $\boxed{\text{ENTER}}$ .
• Enter left hand side of the equation $1$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{1}$ .
• Press $\boxed{\text{WINDOW}}$ key and adjust the window at $\left[-10,20\right]$ by $\left[-10,20\right]$ .
• 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 $10.3188$ years. The approximate value is $10.319$ .

Therefore, the number of years taken to reduce the number of cases to $1000$ is $10.319$ years.

(b)

To determine

To find: The number of years taken to reduce the number of cases to $1$ .

Answer to Problem 32E

The number of years taken to reduce the number of cases to $1$ is $41.275$ years.

Explanation of Solution

Given information: The number of cases are reduced by $20\%$ every year and the number of cases today are $10000$ .

Calculation:

The number of cases is reducing by $20\%$ each year. So, the number of cases reduced to $80\%$ of the total case each year that is the number of cases becomes $0.8$ times every year.

The present number of cases are $10000$ .

Substitute $10000$ for $y_0$ and $0.8$ for $a$ in the exponential function for the number of cases left.

  $\begin{array}{l}y=y_0{a}^t\\ y=10000{\left(0.8\right)}^t\end{array}$

Substitute 1 for $y$ in the exponential function.

  $\begin{array}{c}y=10000{\left(0.8\right)}^t\\ 1=10000{\left(0.8\right)}^t\end{array}$

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 $10000{\left(0.8\right)}^t$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{10000}\boxed{\times }\boxed{(}\boxed{0.8}\boxed{)}\boxed{^}\boxed{\text{X}}$ . Now, press the key $\boxed{\text{ENTER}}$ .
• Enter left hand side of the equation $1$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{1}$ .
• Press $\boxed{\text{WINDOW}}$ key and adjust the window at $\left[-10,80\right]$ by $\left[-10,20\right]$ .
• 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 $41.27540$ years. The approximate value is $41.275$ .

Therefore, the number of years taken to reduce the number of cases to $1$ is $41.275$ years.