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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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## 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 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 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.

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.

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.

Sales Sales of snowblowers are seasonal. Suppose the sales of snowblowers in one region of the country are approximated by

$$ S(t)=500+500 \cos \left(\frac{\pi}{6} t\right) $$

where $t$ is time in months, with $t=0$ corresponding to November. Find the sales for $(a)-(e)$

(a) November $\quad$ (b) January $\quad$ (c) February (d) May (e) August $\quad$ (f) Graph $y=S(t)$

APPLY 11 sales Sales of snowblowers are seasonal.

Suppose the sales of snowblowers in one region of the country

are approximated by

$S(t)=500+500 \cos \left(\frac{\pi}{6} t\right)$

where $t$ is time (in months), with $t=0$ corresponding to

November. The figure below shows a graph of $S .$ Use a definite

integral to find total sales over a year.

Computer sales are generally subject to seasonal fluctuations. An analysis of the sales of a computer manufacturer during $2008-2010$ is approximated by the function

$$s(t)=0.098 \cos ^{2} t+0.387 \quad 1 \leq t \leq 12$$

where $t$ represents time in quarters $(t=1$ represents the end of the first quarter of 2008 ), and $s(t)$ represents computer sales (quarterly revenue) in millions of dollars. Use a double-angle identity to express $s(t)$ in terms of the cosine function.

Seasonal sales: Hank's Heating Oil is a very seasonal enterprise, with sales in the winter far exceeding sales in the summer. Monthly sales for the company can be modeled by $S(x)=1600 \cos \left(\frac{\pi}{6} x-\frac{\pi}{12}\right)+5100,$ where $S(x)$

is the average sales in month $x(x=1 \rightarrow \text { January })$

(a) What is the average sales amount for July?

(b) For what months of the year are sales less than $\$ 4000 ?$

Revenue from Seasonal Merchandise from the sale of air conditioners is seasonal, with the maximum

revenue in summer. Let the revenue (in dollars) received from the sale of air conditioners be approximated by

$R(t)=600 \cos \left(\frac{\pi}{6} t\right)+1000$,

where $t$ is time in months, measured from July 1

$$

\begin{array}{l}{\text { (a) Find } R^{\prime}(t) .} \\ {\text { (b) Find } R^{\prime}(1)} \\ {\text { (c) Find } R^{\prime}(t) \text { for February. }} \\ {\text { (d) Discuss whether the answers in parts (b) and (c) are rea- }} \\ {\text { sonable for this model. }}\end{array}

$$

Toy sales at a department store $t$ months after the month of December can be modeled by the function $s(t)=210+150 \cos \frac{\pi}{6} t,$ where $s$ is in thousands of dollars. What is the value of $s(4),$ and what does it represent? Find the period of this function.

An analysis of sales indicates that demand for a product during a calendar year (no leap year) is modeled by

$$

d(t)=3 \sqrt{t^{2}+1}-2.75 t

$$

where $d$ is demand in thousands of units and $t$ is the day of the year and $t=1$ represents January 1

Economics. Find the average rate of change of the demand of the product over the first quarter.

The rate of sales (in sales per month) of a company is given, for $t$ in months since January $1,$ by

$$r(t)=t^{4}-20 t^{3}+118 t^{2}-180 t+200$$

(a) Graph the rate of sales per month during the first year $(t=0 \text { to } t=12$ ). Does it appear that more sales were made during the first half of the year, or during the second half?

(b) Estimate the total sales during the first 6 months of the year and during the last 6 months of the year.

(c) What are the total sales for the entire year?

(d) Find the average sales per month during the year.

Solve each problem.

Periodic Sales The number of car stereos sold by a national department store chain varies seasonally and is a function of the month of the year. The function

$$x=6.2+3.1 \sin \left(\frac{\pi}{6}(t-9)\right)$$

gives the anticipated sales (in thousands of units) as a function of the number of the month $(t=1,2, \ldots, 12) .$ In what month does the store anticipate selling 9300 units?

Refer to the following:

An analysis of demand $d$ for widgets manufactured by WidgetsRUs (measured in thousands of units per week) indicates that demand can be modeled by the graph below, where $t$ is time in months since January 2010 (note that $t=0$ corresponds to January 2010 ). (GRAPH CAN'T COPY)

Business. Find the period of the graph.

An analysis of sales indicates that demand for a product during a calendar year (no leap year) is modeled by

$$

d(t)=3 \sqrt{t^{2}+1}-2.75 t

$$

where $d$ is demand in thousands of units and $t$ is the day of the year and $t=1$ represents January 1

Economics. Find the average rate of change of the demand of the product over the fourth quarter.

A mathematical model for Merck's sales is given by

$$

S(t)=-0.18 t^{2}+2.5 t+14

$$

where $t$ is time in years and $t=0$ corresponds to 1997.

(A) Compare the model and the data graphically and numerically.

(B) Estimate (to two significant digits) Merck's sales in 2006 and in 2008.

(C) Write a brief verbal description of Merck's sales from 1997

to 2005.

SALES A company that produces snowboards, which are seasonal products, forecasts monthly sales over the next 2 years to be $S = 23.1 + 0.44t + 4.3\ cos(\pi t/6)$, where $S$ is measured in thousands of units and $t$ is the time in months, with $t=1$ representing January 2010. Predict sales for each of the following months.

(a) February 2010

(b) February 2011

(c) June 2010

(d) June 2011

In Exercises 119 and 120 refer to the following:

An analysis of sales indicates that demand for a product during a calendar year is modeled by

$$d=3 \sqrt{t+1}-0.75 t$$

where $d$ is demand in millions of units and $t$ is the month of the year where $t=0$ represents January.

Economics. During which month(s) is demand 3 million units?

The sales $ S $ (in thousands of units) of a new $ CD $ burner after it has been on the market for years are modeled by $ S(t) = 100\left(1 - e^{kt}\right) $. Fifteen thousand units of the new product were sold the first year.

(a) Complete the model by solving for $ k $.

(b) Sketch the graph of the model.

(c) Use the model to estimate the number of units sold after $ 5 $ years.

Refer to the following:

An analysis of demand $d$ for widgets manufactured by WidgetsRUs (measured in thousands of units per week) indicates that demand can be modeled by the graph below, where $t$ is time in months since January 2010 (note that $t=0$ corresponds to January 2010 ). (GRAPH CAN'T COPY)

Business. Find the amplitude of the graph.

Suppose that the ticket sales of an airline (in thousands of dollars) is given by $s(t)=110+2 t+15 \sin \left(\frac{1}{6} \pi t\right),$ where $t$ is measured in months. What real-world phenomenon might cause the fluctuation in ticket sales modeled by the sine term? Based on your answer, what month corresponds to $t=0 ?$ Disregarding seasonal fluctuations, by what amount is the airline's sales increasing annually?

In Exercises 119 and 120 refer to the following:

An analysis of sales indicates that demand for a product during a calendar year is modeled by

$$d=3 \sqrt{t+1}-0.75 t$$

where $d$ is demand in millions of units and $t$ is the month of the year where $t=0$ represents January.

Economics. During which month(s) is demand 4 million units?

DATA ANALYSIS The table shows the average sales $S$ (in millions of dollars) of an outerwear manufacturer for each month $t$, where $t=1$ represents January.

(a) Create a scatter plot of the data.

(b) Find a trigonometric model that fits the data. Graph the model with your scatter plot. How well does the model fit the data?

(c) What is the period of the model? Do you think it is reasonable given the context? Explain your reasoning.

(d) Interpret the meaning of the model's amplitude in the context of the problem.

Monthly Sales Owing to startup costs and seasonal variations, Gina found that the monthly profit in her bagel shop during the first year followed an up-and-down pattern that could be modeled by $P=2 t-7 \sin (\pi t / 3),$ where $P$ was measured in hundreds of dollars and $t$ was measured in months

after January 1.

(a) In what month did the shop first begin to make money?

(b) In what month did the shop enjoy its greatest profit in that

first year?

SALES The following are the slopes of lines representing annual sales $x$ in terms of time in years.Use the slopes to interpret any change in annual sales for a one-year increase in time.

(a) The line has a slope of $m=135$.

(b) The line has a slope of $m=0$.

(c) The line has a slope of $m= -40$.

Refer to the following:

An analysis of demand $d$ for widgets manufactured by WidgetsRUs (measured in thousands of units per weck) indicates that demand can be modeled by the graph below, where $t$ is time in months since January 2010 (note that $t=0$ corresponds to January 2010 ).

GRAPH CANT COPY

Find the period of the graph.

Inventory Management The number of units in inventory in a small company is given by

$$N(t)=25\left(2\left[\frac{t+2}{2}\right]-t\right)$$

where $t$ is the time in months. Sketch the graph of this function and discuss its continuity. How often must this company replenish its inventory?

Sales The rate of change in sales $S$ is inversely proportional to time $t(t>1),$ measured in weeks. Find $S$ as a function of $t$ when the sales after 2 and 4 weeks are 200 units and 300 units, respectively.

Sales A company introduces a new product for which the number of units sold $S$ is $S(t)=200\left(5-\frac{9}{2+t}\right)$ where $t$ is the time in months.

(a) Find the average rate of change of $S$ during the first year.

(b) During what month of the first year does $S^{\prime}(t)$ equal the average rate of change?

Revenue from Seasonal Merchandise from the sale of electric heaters is seasonal with the maximum

revenue in winter. Let the revenue (in dollars) received from the sale of heaters by approximated by

$$R(t)=100 \sin \left(\frac{\pi}{12} t+0.5\right)+400$$

$$\begin{array}{l}{\text { where } t \text { is time in months and } t=0 \text { corresponds to September. }} \\ {\text { (a) Find } R^{\prime}(t) \text { . }} \\ {\text { (b) Find and interpret } R^{\prime}(2)} \\ {\text { (c) Find and interpret } R^{\prime}(5)}\end{array}$$

Refer to the following:

An analysis of demand $d$ for widgets manufactured by WidgetsRUs (measured in thousands of units per weck) indicates that demand can be modeled by the graph below, where $t$ is time in months since January 2010 (note that $t=0$ corresponds to January 2010 ).

GRAPH CANT COPY

Find the amplitude of the graph.

A simple random sample of 5 months of sales data provided the following information:

$\begin{array}{llllll}{\text {Month:}} & {1} & {2} & {3} & {4} & {5} \\ {\text {Units Sold: }} & {94} & {100} & {85} & {94} & {92}\end{array}$

a. Develop a point estimate of the population mean number of units sold per month.

b. Develop a point estimate of the population standard deviation.

a. Business The spreadsheet shows the monthly revenue and monthly expenses for a new business. Find a linear model for monthly revenue and a linear model for monthly expenses.

b. Use the models to predict the month in which revenue will equal expenses.

Use the following information about videocassette sales from 1987 to 1996, where t is the number of years since 1987. The number of blank videocassettes B sold annually in the United States can be modeled by B 15t 281, where B is measured in millions. The wholesale price P for a videocassette can be modeled by P 0.21t 3.52, where P is measured in dollars.

What conclusions can you make from your model about the revenue over time?

SALES From 2003 through 2008, the sales $R_1$ (in thousands of dollars) for one of two restaurants owned by the same parent company can be modeled by

$R_1 = 480 - 8t - 0.8t^2$, $t = 3, 4, 5, 6, 7, 8$

where $t = 3$ represents 2003. During the same six-year period, the sales $R_2$ (in thousands of dollars) for the second restaurant can be modeled by

$R_2 = 254 - 0.78t$, $t = 3, 4, 5, 6, 7, 8$.

(a) Write a function $R_3$ that represents the total sales of the two restaurants owned by the same parent company.

(b) Use a graphing utility to graph $R_1$, $R_2$, and $R_3$ in the same viewing window.

The sales $S$ (in thousands of units) of a new mp3 player after it has been on the market for $t$ years can be modeled by

$$

S(t)=750\left(1-e^{-t t}\right)

$$

a. If 350,000 units of the mp3 player were sold in the first year, find $k$ to four decimal places.

b. Use the model found in part (a) to estimate the sales of the mp3 player after it has been on the market for

3 years.

REVENUE The revenue $R$ (in dollars) generated by the sale of $x$ units of a patio furniture set is given by

$(x-106)^2 = -\dfrac{4}{5}(R-14,045)$.

Use a graphing utility to graph the function and approximate the number of sales that will maximize revenue.

The number of sales per month, $S$, is a function of the amount, $a$ (in dollars), spent on advertising that month, so $S=f(a)$

(a) Interpret the statement $f(1000)=3500$

(b) Which of the graphs in Figure 1.13 is more likely to represent this function?

(c) What does the vertical intercept of the graph of this function represent, in terms of sales and advertising?

(FIGURE CAN'T COPY)

VEHICLE TECHNOLOGY SALES The estimated revenues $r$ (in millions of dollars) from sales of in-vehicle technologies in the United States from 2003 through 2008 can be approximated by the model

$r = 157.30t^2 + 397.4t + 6114, $3 \leq t \leq 8$

where $t$ represents the year, with $t=$ corresponding to 2003.(Source: Consumer Electronics Association)

(a) Use a graphing utility to graph the model.

(b) Find the average rate of change of the model from 2003 through 2008. Interpret your answer in the context of the problem.

Sales Often sales of a new product grow rapidly at first and then level off with time. This is the case with the sales represented by the function

$$S(t)=100-100 t^{-1}$$

where $t$ represents time in years. Find the rate of change of sales for the following numbers of years.

a. 1

b. 10

Alex is a sales representative and earns a base salary of $\$ 1000$ per month plus a $4 \%$ commission on his sales for

the month.

a. Write a linear equation that expresses Alex's monthly salary $y$ in terms of his sales $x .$

b. Graph the equation.

c. What is the $y$ -intercept and what does it represent in the context of this problem?

d. What is the slope of the line and what does it represent in the context of this problem?

e. How much will Alex make if his sales for a given month are $\$ 30,000 ?$

Figure 5.55 shows the number of sales per month made by two salespeople. Which person has the most total sales after 6 months? After the first year? At approximately what times (if any) have they sold roughly equal total amounts? Approximately how many total sales has each person made at the end of the first year?

(FIGURE CAN'T COPY)

Manufacturing Swimwear Get Wet, Inc., manufactures swimwear, a seasonal product. The monthly sales $x$ (in thousands) for Get Wet swimsuits are modeled by the equation

$$x=72.4+61.7 \sin \frac{\pi t}{6},$$

where $t=1$ represents January, $t=2$ February, and so on. Estimate the number of Get Wet swimsuits sold in January, April, June, October, and December. For which two of these months are sales the same? Explain why this might be so.

Sales Sales of a new model of compact disc player are approximated by the function $S(x)=1000-800 e^{-x},$ where $S(x)$ is in appropriate units and $x$ represents the number of years the disc player has been on the market.

a. Find the sales during year $0 .$

b. In how many years will sales reach 500 units?

c. Will sales ever reach 1000 units?

d. Is there a limit on sales for this product? If so, what is it?

Sales Sales of a new model of compact disc player are approx- imated by the function $S(x)=1000-800 e^{-x},$ where $S(x)$ is in appropriate units and $x$ represents the number of years the

disc player has been on the market.

(a) Find the sales during year 0.

(b) In how many years will sales reach 500 units?

(c) Will sales ever reach 1000 units?

(d) Is there a limit on sales for this product? If so, what

is it?

Sales The sales of a small company were $\$ 13,000$ in its third year of operation and $\$ 37,000$ in its seventh year. Let $y$ represent sales in the $x$ th year of operation. Assume that the data can be approximated by a straight line.

(a) Find the slope of the sales line, and give an equation for the line in the form $y=m x+b .$

(b) Use your answer from part (a) to find out how many years must pass before the sales surpass $\$ 50,000 .$

Research indicates that monthly profit for Widgets R Us is modeled by the function

$$P=-100+(0.2 q-3) q$$

where $P$ is profit measured in millions of dollars and $q$ is the quantity of widgets produced measured in thousands.

Find the break-even point for a month to the nearest unit.

You will use linear functions to study real-world problems.

Sales The number of computers sold per year since 2001 by T.J.'s Computers is given by the linear function $n(t)=25 t+350 .$ Here, $t$ is the number of years since 2001

(a) How many computers were sold in $2005 ?$

(b) What is the $y$ -intercept of this function, and what does it represent?

(c) According to the function, in what year will 600 computers be sold?

Coffee sales fluctuate with the weather, with a great deal more coffee sold in the winter than in the summer. For Joe's Diner, assume the function $G(x)=21 \cos \left(\frac{2 \pi}{365} x+\frac{\pi}{2}\right)+29$ models daily coffee sales (for non-leap years), where $G(x)$ is the number of gallons sold and $x$ represents the days of the year $(x=1 \rightarrow \text { January } 1)$

(a) How many gallons are projected to be sold on March $21 ?$ (b) For what days of the year are more than 40 gal of coffee sold?

[T] The temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function

$$

T(x)=5+18 \sin \left[\frac{\pi}{6}(x-4.6)\right]

$$

where $x$ is time in months and $x=1.00$ corresponds to January $1 .$ Determine the month and day when the temperature is $21^{\circ} \mathrm{C} .$