Chapter 3.4, Problem 109E

a.

To determine

To find:

An interval containing the solution of the equation by completing the table.

Answer to Problem 109E

Our required interval would be $(8,9)$ .

Explanation of Solution

Given:

A logarithmic equation and a table: $20\left(100-e^{x/2}\right)=500$ .

x	5	6	7	8	9
$20\left(100-e^{x/2}\right)$

Calculation:

To complete the table, we will substitute the values of x in our given expression as shown below:

  $\begin{array}{l}20\left(100-e^{x/2}\right)\\ \Rightarrow 20\left(100-e^{5/2}\right)\\ \Rightarrow 1756.3501207\approx 1756.350\end{array}$

  $\begin{array}{l}20\left(100-e^{x/2}\right)\\ \Rightarrow 20\left(100-e^{6/2}\right)\\ \Rightarrow 20\left(100-e^3\right)\\ \Rightarrow 1598.289261\approx 1598.289\end{array}$

  $\begin{array}{l}20\left(100-e^{x/2}\right)\\ \Rightarrow 20\left(100-e^{7/2}\right)\\ \Rightarrow 20\left(100-e^{3.5}\right)\\ \Rightarrow 1337.690960\approx 1337.691\end{array}$

  $\begin{array}{l}20\left(100-e^{x/2}\right)\\ \Rightarrow 20\left(100-e^{8/2}\right)\\ \Rightarrow 20\left(100-e^4\right)\\ \Rightarrow 908.036999\approx 908.037\end{array}$

  $\begin{array}{l}20\left(100-e^{x/2}\right)\\ \Rightarrow 20\left(100-e^{9/2}\right)\\ \Rightarrow 20\left(100-e^{4.5}\right)\\ \Rightarrow 199.6573739\approx 199.657\end{array}$

The complete table would be:

x	5	6	7	8	9
$20\left(100-e^{x/2}\right)$	1756.350	1598.289	1337.691	908.037	199.657

Since 500 lies between 908.037and 199.657, therefore our required interval would be $(8,9)$ .

b.

To determine

To graph:

The given function using a graphing utility and approximate the solution.

Answer to Problem 109E

The solution of our given equation would be $x=8.635$ .

Explanation of Solution

Given:

A logarithmic equation: $20\left(100-e^{x/2}\right)=500$ .

Calculation:

Upon graphing our given function, we will get our required graph as shown below:

  

We can see that both lines intersect at $x=8.635$ , therefore, $x=8.635$ is a solution for our given function.

c.

To determine

To solve:

The given equation algebraically.

Answer to Problem 109E

The solution of our given equation would be $x\approx 8.635$ .

Explanation of Solution

Given:

A logarithmic equation: $20\left(100-e^{x/2}\right)=500$ .

Calculation:

We will use logarithmic properties to solve our given equation.

  $\begin{array}{l}20\left(100-e^{x/2}\right)=500\left(\text{Given equation}\right)\\ 2000-20e^{x/2}=500\left(\text{Using distributive property}\right)\\ 2000-2000-20e^{x/2}=500-2000\left(\text{Subtracting 2000 from both sides}\right)\\ -20e^{x/2}=-1500\\ \frac{-20e^{x/2}}{-20}=\frac{-1500}{-20}\left(\text{Dividing both sides by }-20\right)\\ e^{x/2}=75\\ \ln \left(e^{x/2}\right)=\ln (75)\left(\text{Taking natural log on both sides}\right)\\ \frac{x}{2}\cdot \ln (e)=\ln (75)\left(\text{Using property ln}\left(x^m\right)=m\ln (x)\right)\\ \frac{x}{2}=4.317488113\left(\text{Using property ln(}e\text{)}=\text{1}\right)\\ \frac{x}{2}\cdot 2=4.317488113\cdot 2\left(\text{Multiplying both sides by 2}\right)\\ x=8.634976227\\ x\approx 8.635\left(\text{Rounding to three decimal places}\right)\end{array}$

Therefore, the solution of our given equation would be $x\approx 8.635$ .