Chapter 4.6, Problem 72E

Solution

(a)

To calculate : The company's sales in 2012.

The company's sales in 2012 is $105$ million dollars.

Given information :

The business manager of a small manufacturing company finds that she can model the company's annual growth as roughly exponential but with cyclical fluctuations. She uses the function $S\left(t\right)=105{\left(1.04\right)}^t+4\sin \left(\frac{\pi t}{3}\right)$ to estimate sales (in million of dollars), $t$ years after 2012.

Calculation :

Consider the given function $S\left(t\right)=105{\left(1.04\right)}^t+4\sin \left(\frac{\pi t}{3}\right)$ .

To estimate the company's sales in 2012, substitute $t=0$ .

  $\begin{array}{c}S\left(0\right)=105{\left(1.04\right)}^0+4\sin \left(\frac{\pi \left(0\right)}{3}\right)\\ =105+4\sin 0\\ =105\end{array}$

Therefore, the sales in 2012 is $105$ million dollars.

(b)

To calculate : The approximate annual growth rate.

The approximate annual growth rate is $7.66$ million dollars.

Given information :

The business manager of a small manufacturing company finds that she can model the company's annual growth as roughly exponential but with cyclical fluctuations. She uses the function $S\left(t\right)=105{\left(1.04\right)}^t+4\sin \left(\frac{\pi t}{3}\right)$ to estimate sales (in million of dollars), $t$ years after 2012.

Calculation :

Consider the given function $S\left(t\right)=105{\left(1.04\right)}^t+4\sin \left(\frac{\pi t}{3}\right)$ .

To estimate the growth rate, find the company's sales in 2013 and for this substitute $t=1$ .

  $\begin{array}{c}S\left(0\right)=105{\left(1.04\right)}^1+4\sin \left(\frac{\pi \left(1\right)}{3}\right)\\ =109.2+4\sin \frac{\pi }{3}\\ =109.2+4\left(\frac{\sqrt{3}}{2}\right)\\ =112.66\end{array}$

Thus the growth rate is,

  $\begin{array}{l}=\frac{112.66-105}{1}\\ =7.66\end{array}$

Therefore, the annual growth rate is $7.66$ million dollars.

(c)

To calculate : The sales for 2020.

The sales for 2020 is $147.16$ million dollars.

Given information :

The business manager of a small manufacturing company finds that she can model the company's annual growth as roughly exponential but with cyclical fluctuations. She uses the function $S\left(t\right)=105{\left(1.04\right)}^t+4\sin \left(\frac{\pi t}{3}\right)$ to estimate sales (in million of dollars), $t$ years after 2012.

Calculation :

Consider the given function $S\left(t\right)=105{\left(1.04\right)}^t+4\sin \left(\frac{\pi t}{3}\right)$ .

To find the company's sales in 2020 and for this substitute $t=8$ .

  $\begin{array}{c}S\left(0\right)=105{\left(1.04\right)}^8+4\sin \left(\frac{\pi \left(8\right)}{3}\right)\\ =143.70+4\sin \frac{8\pi }{3}\\ =143.70+3.46\\ =147.16\end{array}$

Therefore, the sales in 2020 is $147.16$ million dollars.

(d)

To calculate : The number of years in each economic cycle.

The number of years in each economic cycle is $6$ .

Given information :

The business manager of a small manufacturing company finds that she can model the company's annual growth as roughly exponential but with cyclical fluctuations. She uses the function $S\left(t\right)=105{\left(1.04\right)}^t+4\sin \left(\frac{\pi t}{3}\right)$ to estimate sales (in million of dollars), $t$ years after 2012.

Calculation :

Consider the given function $S\left(t\right)=105{\left(1.04\right)}^t+4\sin \left(\frac{\pi t}{3}\right)$ .

The period of the model function is same as $\sin \left(\frac{\pi t}{3}\right)$ .

The period of the function $a\sin \left(bx+c\right)+d$ is $\frac{2\pi }{\left|b\right|}$ .

So, the period of $\sin \left(\frac{\pi t}{3}\right)$ is $\frac{2\pi }{\frac{\pi }{3}}=6$ .

Therefore, the number of years in each economic cycle is $6$ .