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Brookdale Community College *
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156
Subject
Mathematics
Date
Apr 3, 2024
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4
Uploaded by DeaconEchidnaPerson1035
8/2012 MATH 156 Unit 1 Project DUE DATE: All projects must be TYPED and STAPLED. PART 1: ONLINE BOOK STORE
1
Sean O’Hagan is CEO of SOBooks.com, an online bookstore. Since th
e establishment of the company website six years ago, the number of visitors to the site has grown quite dramatically, as indicated by the following table: a)
Graph the function V(t)
as a function of time
t
. b)
What type of function can be described by your graph? c)
Predict Web site traffic in year 8. As the online bookstore, SOBooks.com, has grown in popularity, the sales manager has been monitoring book sales as a function of the traffic at your site (measured in “hits” per day) and has obtained the following model. 0.04 if 0
1,000
( )
0.045
9 if 1,000
2,000
x
x
n x
x
x
=
−
Where n(x) is the average number of books sold in a day in which there are x hits at the site. d)
On average, how many books per day does the model predict the bookstore will sell when it has 500 hits in a day? 1,000 hits in a day? 1,500 hits in a day? e)
What does the coefficient 0.045 tell you about book sales? f)
According to the model, how many hits per day will be needed in order to sell an average of 30 books per day? 1
Adapted from Applications in finite Mathematics+applied Calc
by Waner/Costenoble, Cengage, 2011 Year t
0 1 2 3 4 5 6 Website Traffic V(t)
(visits/day) 100 725 1350 1975 2600 3225 3850
To increase business at SOBooks.com, Sean plans to place more banner ads at well-known Internet portals. So far, he has the following data on the average number of hits per day at SOBooks.com versus monthly advertising expenditures. He decides to construct a linear model giving the average number of hits h
per day as a function of the advertising expenditure c
. g)
What is the model he obtains? h)
Based on your model, how much traffic can he anticipate if he budgets $8,000 per month for banner ads? i)
The goal is to eventually increase traffic at the site to an average of 3,000 hits per day. Based on the model, how much should be spent on banner ads in order to accomplish this? A month ago Sean O’Hagan increased expenditure on banner ads to $8,000 per month, and noticed that the traffic at SOBooks.com has not increased to the level predicted by the linear model in parts g) –
i). Fitting a quadratic function to the available data gives the model 2
0.000005
0.085
1950
h
c
c
= −
+
+
where h is the daily traffic (hits) at the Web site, and c is the monthly advertising expenditure. j)
According to this model, what is the current traffic at the site? k)
Does this model give a reasonable prediction of traffic at expenditures larger than $8,500 per month? Why?
Besides selling books, SOBooks.com is generating revenue through its new online publishing service. Readers pay a fee to download the entire text of a novel. Author royalties and copyright fees cost you an average of $3.25 per novel, and the monthly cost of operating and maintaining the service amounts to $900 per month. The company is currently charging readers $5.00 per novel. L)
What are the associated cost, revenue, and profit functions? m)
How many novels must be sold per month in order to break even? What is the minimum number of novels that must be sold in order to turn a profit? Advertising Expenditure ($/Month) $2,000 $6,000 Website Traffic (Hits/Day) 1,900 2,500
n)
If the charge is lowered to $4.50 per novel, how many books must be sold in order to break even? In order to generate a profit from its online publishing service, SOBooks.com needs to know how the demand for novels depends on the price it charges. During the first month of the service, it was charging $10 per novel, and sold 350. Lowering the price to $5.00 per novel had the effect of increasing demand to 620 novels per month. o)
Use the given data to construct a linear demand equation. p)
Use the demand equation you constructed in part o) to estimate the demand if you raised the price to $15 per novel. Round your answer to the nearest number q)
Using the information on cost, revenue, and profit, given in parts L) and O),
determine which of the two prices ($5 and $10) would result in the largest profit
, and the size of that profit
. It is now several months later, and SOBooks.com has tried selling its online novels at a variety of prices, with the following result. r)
Use the given data to obtain a linear regression model of demand. (Round coefficients to four decimal places.) In the model, demand is the independent variable and price is the dependent variable. s)
Use the demand model you constructed in part (r) to estimate the demand if the company charged $8 per novel. (Round the answer to the nearest novel.) Part 2: The Good Buy Company
The Good Buy Company manufactures DVD players. The marketing department finds that if the company manufactures x DVD players per days its costs are ( )
150
10000
C x
x
=
+
and its revenues are 2
( )
2
600
R x
x
x
= −
+
. Price $5.00 $10 $12 $15 Demand (Montly Sales) 620 350 300 100
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a.
On the same set of axis graph both functions and clearly label each function. State the window used. b.
Find the break-even points, algebraically, graphically, or numerically and show work to support your answer. State your solution in a complete sentence. c.
How many DVD players should Good Buy produce in order to make money each day? d.
How many DVD players should the firm make in order to maximize revenue? e.
What is the equation for the profit function? f.
How many DVD players should the firm make in order to maximize profit?