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Oregon State University

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Problem 75

Sales The monthly sales $S$ (in thousands of units) of a seasonal product are modeled by

$$S=58.3+32.5 \cos \frac{\pi t}{6}$$

where $t$ is the time (in months), with $t=1$ corresponding to January. Use a graphing utility to graph the model for $S$ and determine the months when sales exceed $75,000$ units.

Answer

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## Discussion

## Video Transcript

All right, let's using graphing utility for this problem. So here's my graphing calculator, and you can go into the y equals menu and type the function in there. So this is the function that represents the sales and thousands of units of the seasonal product. And I'm using X in my calculator in place of tea, and that would be the time in months. So X equals one is going to be January. So here's what the graph looks like. And for my window dimensions, I went from 0 to 12 for the months and then 0 to 140 for the units in thousands. Okay, so we're interested in knowing which months have sales greater than 100,000. So I'm going to go back to my Y, equals menu and go toe white, too, and put 100 there, which represents 100,000 and that's going to give us a horizontal line, and we want to know which months go above that line. So let's find those intersection points. Roughly. What we can do is move that cursor along, so this intersection point is at about 4.9, and then on the other side. This intersection point is at about 1.2. So then months 23 and four fallen between month to would be February 3 would be march and for would be April, so sales exceed 100,000 units in February, March and April.

## Recommended Questions

The monthly sales $ S $ (in thousands of units) of a seasonal product are approximated by

$ S = 74.50 + 43.75 \sin \dfrac{\pi t}{6} $

where $ t $ is the time (in months), with $ t = 1 $ corresponding to January. Determine the months in which sales exceed $ 100,000 $ units.

SALES The projected monthly sales $S$ (in thousands of units) of lawn mowers (a seasonal product) are modeled by $S =\ 74\ +\ 3t\ -\ 40\ cos(\pi t/6)$, where $t$ is the time (in months), with $t=1$ corresponding to January. Graph the sales function over 1 year.

Sales The sales $S$ (in thousands of units) of a seasonal product are given by the model

$$S=74.50+43.75 \sin \frac{\pi t}{6}$$

where $t$ is the time in months, with $t=1$ corresponding to January. Find the average sales for each time period.

(a) The first quarter $(0 \leq t \leq 3)$

(b) The second quarter $(3 \leq t \leq 6)$

(c) The entire year $(0 \leq t \leq 12)$

Sales A company that produces snowboards, which are seasonal products, forecasts monthly sales for 1 year to be

$S=74.50+43.75 \cos \frac{\pi t}{6}$

where $S$ is the sales in thousands of units and $t$ is the time in months, with $t=1$ corresponding to January.

(a) Use a graphing utility to graph the sales function over the one-year period.

(b) Use the graph in part (a) to determine the months of maximum and minimum sales.

The monthly sales $ S $(in hundreds of units) of skiing equipment at a sports store are approximated by

$ S = 58.3 + 32.5 \cos \dfrac{\pi t}{6} $

where $ t $ is the time (in months), with $ t = 1 $ corresponding to January. Determine the months in which sales exceed $ 7500 $ units.

The monthly unit sales $U$ (in thousands) of lawn mowers are approximated by

$$U=74.50-43.75 \cos \frac{\pi t}{6}$$where $t$ is the time (in months), with $t=1$ corresponding to January. Determine the months in which unit sales exceed 100,000

Annual sales of a product are generally subject to seasonal fluctuations and are approximated by the function

$$s(t)=4.3 \cos \left(\frac{\pi}{6} t\right)+56.2 \quad 0 \leq t \leq 11$$

where $t$ represents time in months $(t=0$ represents January) and $s(i)$ represents monthly sales of the product in thousands of dollars.

Find the month(s) in which monthly sales are 51,900 dollars.

Annual sales of a product are generally subject to seasonal fluctuations and are approximated by the function

$$s(t)=4.3 \cos \left(\frac{\pi}{6} t\right)+56.2 \quad 0 \leq t \leq 11$$

where $t$ represents time in months $(t=0$ represents January) and $s(i)$ represents monthly sales of the product in thousands of dollars.

Find the month(s) in which monthly sales are 56,200 dollars.

The monthly unit sales $U$ (in hundreds) of skis for a chain of sports stores are approximated by

$U=58.3+32.5 \cos \frac{\pi t}{6}$

where $t$ is the time (in months), with $t=1$ corresponding to January. Determine the months in which unit sales exceed 7500

A company that produces snowboards, which are seasonal products, forecasts monthly sales for one year to be $$S=74.50+43.75 \cos \frac{\pi t}{6}$$ where $S$ is the sales in thousands of units and $t$ is the time in months, with $t=1$ corresponding to January.

(a) Use a graphing utility to graph the sales function over the one-year period.

(b) Use the graph in part (a) to determine the months of maximum and minimum sales.