Chapter 4.4, Problem 100E

a.

To determine

Predict the sales in the month of February $2014$ .

Answer to Problem 100E

  $\boxed{26,134units}$

Explanation of Solution

Given information:

A company that produces snowboards forecasts monthly sales over the next $2$ years to be $S=23.1+0.442t+4.3\cos \frac{\pi t}{6}$ where $S$ is measured in thousands of units and $t$ is the time in months, with $t=1$ representing January $2014$ . Predict sales for each of the following months.

February $2014$

Calculation:

Monthly sales forecasted for over $2$ years from the producers of snowboards is given by,

  $S=23.1+0.442t+4.3\cos \frac{\pi t}{6}$

Where, $t$ is time in months with $t=1$ representing January $2014$ .

It follows that for February $2014$ , $t=2$ , it gives

  $S=23.1+0.442\left(2\right)+4.3\cos \frac{\pi \left(2\right)}{6}$

  $\begin{array}{l}=23.1+0.884+4.3\cos \frac{\pi }{3}\\ =23.984+4.3\left(0.5\right)\\ =23.984+2.15\end{array}$

That is,

  $S=26.134$

Hence, sales for the month February $2014$ would be $\boxed{26,134units}$

b.

To determine

Predict the sales in the month of February $2015$ .

Answer to Problem 100E

  $\boxed{31,438units}$

Explanation of Solution

Given information:

A company that produces snowboards forecasts monthly sales over the next $2$ years to be $S=23.1+0.442t+4.3\cos \frac{\pi t}{6}$ where $S$ is measured in thousands of units and $t$ is the time in months, with $t=1$ representing January $2014$ . Predict sales for each of the following months.

February $2015$

Calculation:

Monthly sales forecasted for over $2$ years from the producers of snowboards is given by,

  $S=23.1+0.442t+4.3\cos \frac{\pi t}{6}$

Where, $t$ is time in months with $t=1$ representing January $2014$ .

It follows that for February $2015$ , $t=12+2=14$ , it gives

  $S=23.1+0.442\left(14\right)+4.3\cos \frac{\pi \left(14\right)}{6}$

  $=23.1+6.188+4.3\cos \frac{7\pi }{3}$

Because, $\cos \frac{7\pi }{3}=\cos \left(2\pi +\frac{\pi }{3}\right)=\cos \frac{\pi }{3}$ , then

  $S=29.288+4.3\left(0.5\right)$

  $=29.288+2.15$

  $=31.438$

Hence, sales for moth February $2015$ would be $\boxed{31,438units}$ .

c.

To determine

Predict the sales in the month of June $2014$ .

Answer to Problem 100E

  $\boxed{21,452units}$

Explanation of Solution

Given information:

A company that produces snowboards forecasts monthly sales over the next $2$ years to be $S=23.1+0.442t+4.3\cos \frac{\pi t}{6}$ where $S$ is measured in thousands of units and $t$ is the time in months, with $t=1$ representing January $2014$ . Predict sales for each of the following months.

June $2014$

Calculation:

Monthly sales forecasted for over $2$ years from the producers of snowboards is given by,

  $S=23.1+0.442t+4.3\cos \frac{\pi t}{6}$

Where, $t$ is time in months with $t=1$ representing January $2014$ .

It follows that for June $2014$ , $t=6$ , it gives

  $S=23.1+0.442\left(6\right)+4.3\cos \frac{\pi \left(6\right)}{6}$

  $\begin{array}{l}=23.1+2.652+4.3\cos \pi \\ =25.752+4.3\left(-1\right)\\ =25.752-4.3\end{array}$

That is

  $S=21.452$

Hence, sales for the moth June $2014$ would be $\boxed{21,452units}$

d.

To determine

Predict the sales in the month of June $2015$ .

Answer to Problem 100E

  $\boxed{26,756units}$

Explanation of Solution

Given information:

A company that produces snowboards forecasts monthly sales over the next $2$ years to be $S=23.1+0.442t+4.3\cos \frac{\pi t}{6}$ where $S$ is measured in thousands of units and $t$ is the time in months, with $t=1$ representing January $2014$ . Predict sales for each of the following months.

June $2015$

Calculation:

Monthly sales forecasted for over $2$ years from the producers of snowboards is given by,

  $S=23.1+0.442t+4.3\cos \frac{\pi t}{6}$

Where, $t$ is time in months with $t=1$ representing January $2014$ .

It follows that for June $2015$ , $t=12+6=18$ , it gives

  $S=23.1+0.442\left(18\right)+4.3\cos \frac{\pi \left(18\right)}{6}$

  $=23.1+7.956+4.3\cos 3\pi$

Because , $\cos 3\pi =\cos \left(2\pi +\pi \right)=\cos \pi$ then,

  $\begin{array}{l}S=31.056+4.3\left(-1\right)\\ =31.056-4.3\\ =26.756\end{array}$

Hence, sales for the month June $2015$ would be $\boxed{26,756units}$