Chapter 11.5, Problem 60E

(a)

To determine

To Find: The annual sales of the product after 9 years.

Answer to Problem 60E

The sale is of $91116$ products.

Explanation of Solution

Given:

After t years, the annual sales in hundreds of thousands of units of a product $q$is given by $q={\left(\frac{1}{2}\right)}^{{0.8}^t}$

Calculation:

Consider the given expression is,

  $q={\left(\frac{1}{2}\right)}^{{0.8}^t}$

Then,

  $\begin{array}{c}q={\left(\frac{1}{2}\right)}^{{0.8}^9}\left({10}^5\right)\\ =91116.4\\ \cong 91116\end{array}$

Thus, the sale is of $91116$ .

(b)

To determine

To Find: The time after which the the annual sales is about 95,350 units

Answer to Problem 60E

The time required is of $0.085\text{years}$ .

Explanation of Solution

Consider the given expression is,

  $q={\left(\frac{1}{2}\right)}^{{0.8}^t}$

Then,

  $\begin{array}{l}{\left(\frac{1}{2}\right)}^{{0.8}^t}=0.95350\\ \frac{1}{2^{0.8t}}=0.95350\\ \ln 2^{-0.8t}=\ln 0.95350\\ -0.8t\ln 2=\ln 0.95350\end{array}$

Solve further as,

  $\begin{array}{c}t=\frac{\ln 0.95350}{-0.8\ln 2}\\ =0.0858689\\ \cong 0.085\text{years}\end{array}$