Chapter 4.4, Problem 124E

a.

To determine

To estimate: sales for January 2014.

Answer to Problem 124E

  $S\approx 0.9368\text{hundreds}\text{of}\text{unit}\text{.}$

Explanation of Solution

Given information: A company that produces wakeboards forecasts monthly sales S

over two-year period to be $S=2.7+0.142t+2.2\sin (\frac{\pi t}{6}-\frac{\pi }{2}).$Where S is measured in hundreds of units and t is the time (in months),with t =1 corresponding to January 2014.

Formula used:

Monthly sales S over two-year period to be $S=2.7+0.142t+2.2\sin (\frac{\pi t}{6}-\frac{\pi }{2}).$

Calculation:

To estimate sales for January 2014 substitute t =1 in the monthly sales equation.

  $\begin{array}{l}S=2.7+0.142t+2.2\sin (\frac{\pi t}{6}-\frac{\pi }{2}).\\ \text{Substitute}t=1.\\ S=2.7+0.142\times 1+2.2\sin (\frac{\pi \times 1}{6}-\frac{\pi }{2})\\ S=2.7+0.142+2.2\sin (\frac{\pi -3\pi }{6})\\ S=2.7+0.142+2.2\sin (-\frac{\pi }{3}).\\ S\approx 2.7+0.142+2.2\times (-0.866).\\ S\approx 0.9368\text{hundreds}\text{of}\text{unit}\text{.}\end{array}$

b.

To determine

To estimate: sales for February 2015.

Answer to Problem 124E

  $S\approx 3.588\text{ hundreds}\text{of}\text{unit}\text{.}$

Explanation of Solution

Given information: A company that produces wakeboards forecasts monthly sales S

over two-year period to be $S=2.7+0.142t+2.2\sin (\frac{\pi t}{6}-\frac{\pi }{2}).$Where S is measured in hundreds of units and t is the time (in months), with t =1 corresponding to January 2014.

Formula used:

Monthly sales S over two-year period to be $S=2.7+0.142t+2.2\sin (\frac{\pi t}{6}-\frac{\pi }{2}).$

Calculation:

To estimate sales for February 2015 substitute t =14 in the monthly sales equation because February 2015 is 14 months after January 2014.

  $\begin{array}{l}S=2.7+0.142t+2.2\sin (\frac{\pi t}{6}-\frac{\pi }{2}).\\ \text{Substitute}t=14.\\ S=2.7+0.142\times 14+2.2\sin (\frac{\pi \times 14}{6}-\frac{\pi }{2})\\ S=2.7+1.988+2.2\sin (\frac{11\pi }{6})\\ S\approx 3.588\text{ hundreds}\text{of}\text{unit}\text{.}\end{array}$

c.

To determine

To estimate: sales for May 2014.

Answer to Problem 124E

  $S\approx 5.315\text{ hundreds}\text{of}\text{unit}\text{.}$

Explanation of Solution

Given information: A company that produces wakeboards forecasts monthly sales S

over two-year period to be $S=2.7+0.142t+2.2\sin (\frac{\pi t}{6}-\frac{\pi }{2}).$Where S is measured in hundreds of units and tis the time (in months),with t =1 corresponding to January 2014.

Formula used:

Monthly sales S over two-year period to be $S=2.7+0.142t+2.2\sin (\frac{\pi t}{6}-\frac{\pi }{2}).$

Calculation:

To estimate sales for May 2014 substitute t =5 in the monthly sales equation because May 2014 is 5 months after January 2014.

  $\begin{array}{l}S=2.7+0.142t+2.2\sin (\frac{\pi t}{6}-\frac{\pi }{2}).\\ \text{Substitute}t=5.\\ S=2.7+0.142\times 5+2.2\sin (\frac{\pi \times 5}{6}-\frac{\pi }{2})\\ S=2.7+0.71+2.2\sin (\frac{\pi }{3}).\\ S\approx 5.315\text{ hundreds}\text{of}\text{unit}\text{.}\end{array}$

d.

To determine

To estimate: sales for June 2015.

Answer to Problem 124E

  $S\approx 7.456\text{ hundreds}\text{of}\text{unit}\text{.}$

Explanation of Solution

Given information: A company that produces wakeboards forecasts monthly sales S

over two-year period to be $S=2.7+0.142t+2.2\sin (\frac{\pi t}{6}-\frac{\pi }{2}).$Where S is measured in hundreds of units and tis the time (in months),with t =1 corresponding to January 2014.

Formula used:

Monthly sales S over two-year period to be $S=2.7+0.142t+2.2\sin (\frac{\pi t}{6}-\frac{\pi }{2}).$

Calculation:

To estimate sales for June 2015 substitute t =18 in the monthly sales equation because June 2015 is 18 months after January 2014.

  $\begin{array}{l}S=2.7+0.142t+2.2\sin (\frac{\pi t}{6}-\frac{\pi }{2}).\\ \text{Substitute}t=18.\\ S=2.7+0.142\times 18+2.2\sin (\frac{\pi \times 18}{6}-\frac{\pi }{2}).\\ S=2.7+2.556+2.2\sin (\frac{5\pi }{2}).\\ S\approx 7.456\text{ hundreds}\text{of}\text{unit}\text{.}\end{array}$