 # Math X 410 Business Applications of Calculus

The problem set including the extra credit question is as follows:

PART I

As manager of a particular product line, you have data available for the past 11 sales periods. This data associates your product line’s units sold “x” and total PROFIT “P” results for these sales periods.

Product Red03
Units [x] Profit [P]
10 -33986
20 -31792
100 -9200
130 790
190 21418
240 37728
300 54000
320 58208
380 65840
430 65050
500 50000

Section A: 1st Order Model

1.  Use Microsoft Excel’s Chart feature to graph a plot of the data,
assuming P = ƒ(x). Add the most appropriate 1st order “trend line”, the
equation of this line, and the equation’s coefficient of determination—its
“[(R2)]”.

2. Answer the following questions using this 1st order model. Assume that, unless otherwise indicated, the restricted domain for “x” is 0 = x = 510 units.

a.  Estimate Profit “P” @ “x” = 0 units and “x” = 70 units.

b.  Estimate how many units “x” of the product must be sold in order
to generate a PROFIT of \$0.00 and a PROFIT of \$35,000.

c.  Calculate how many product units “x” should be sold per sales
period to optimize this product’s PROFIT “P” and the value of “P” at
this “x” value. Assume market constraints suggest the maximum
number of product units that actually can be sold per sales period may
not exceed…
(1). …510 (0 = x = 510 units). (2). …300 (0 = x = 300 units).

d.  Estimate marginal PROFIT “mP” for this product if initially…
(1). …480 units were sold. (2). …300 units were sold.

Section B: 2nd Order Model

1.  Use Microsoft Excel’s Chart feature to graph a plot of the data,
assuming P = ƒ(x). Add the most appropriate 2nd order “trend line”, the
equation of this line, the equation’s coefficient of determination—its

2. Answer the following questions using this 2nd order model. Assume that, unless otherwise indicated, the restricted domain for “x” is 0 = x = 510 units.

a.  Estimate Profit “P” @ “x” = 0 units and “x” = 70 units.

b.  Estimate how many units “x” of the product must be sold in order
to generate a PROFIT of \$0.00 and sold in a PROFIT of \$35,000.

c.  Calculate how many product units “x” should be sold per sales
period to optimize this product’s PROFIT “P” and the value of “P” at
this “x” value. Assume market constraints suggest the maximum
number of product units that actually can be sold per sales period may
not exceed…
(1). …510 (0 = x = 510 units). (2). …300 (0 = x = 300 units).

d.  Use differential calculus to provide an estimate of marginal
PROFIT “mP” for this product if initially…
(1). …480 units were sold. (2). …300 units were sold.

Section C: The Most Appropriate Model

1.  Identify which of the two PROFIT models derived above—1st or 2nd
order—is most appropriate for estimating purposes, according to the
“highest percent variation explained” criterion—a criterion based on
[(R2)] or [(R2)adj]. Based on which of the two models you feel is most
appropriate, would you say that the results for the 1st order or 2nd order
model are most realistic?

PART II

As manager of product line Blue03, you have the following data available for the past 6 sales periods. This data associates your product line’s demand (units sold) “x” and unit price “p” results for these sales periods.

Product Blue03
Demand [x] Price [p]
200 800
400 900
600 500
1200 600
1600 150
2000 50

Section A: DEMAND Model Development

1. 1st Order: Use the Chart feature of Microsoft Excel® to help derive…
a.  …the product’s “best fitting” 1st order model p = ƒ(x).
b.  …the model’s coefficient of determination “[(R2)]”. Then,
interpret the “[(R2)]” value.

2. 2nd Order: If the “[(R2)]” value of the 1st order model is not “+1”, use
the Chart feature of Microsoft Excel® to help derive…
a.  …the “best fitting” polynomial, 2nd order model p = ƒ(x).
b.  …identify the model’s coefficient of determination “[(R2)]”, and

Section B: Developing the Models to be Used in Subsequent Analyses

1.  DEMAND. Identify which of the DEMAND models derived
above—1st order or 2nd order—best meets our course’s “highest percent
variation explained” criterion. Use this model to answer the questions
that follow.

2.  REVENUE. Create the REVENUE model R = f(x) from the
DEMAND model identified in “1” above.

3.  COST, REVENUE and PROFIT. Assume you had comparable
COST “C” and units produced “x” data for the same 6 sales periods, and,
after using Excel’s Chart feature to develop 1st and 2nd order “trend line”
equations and appropriate “[(R2)]” values, you selected the 2nd order
equation C = –0.1515(x)2 + 345.01(x) + 137559 to use in further
analyses. Create the PROFIT model P = f(x) from the COST model and
from the REVENUE model identified in “2” above.

Section C: “Break Even”, Optimization and Advanced Topics [0 = x = 1,100 units]

1.  Calculate how many product units “x” must be produced and sold in
order to generate a PROFIT of \$0.00. Assume market constraints are
currently such that “x” cannot exceed 1,100 units per sales period.

2.  Determine “C” and “R” at the quantity “x” where “P” = \$0.00.

3. Differential calculus may be used as part of a process to develop
optimization estimates for “Rmax” and “Pmax”. Based on the market
constraints shown below, calculate the number of product units “x” that
should be sold per sales period to maximize REVENUE and PROFIT
…then…calculate “Rmax” and “Pmax” at these “x” values.
a.  1,100 units (0 = x = 1,100). b.  …850 units (0 = x = 850).

4. Determine the unit price “p” that should be charged per sales period to
optimize this product’s “R” and “P” based on the constraints of…
a.  …3a above (0 = x = 1,100 units). b.  …3b above (0 = x = 850 units).

5.  Using your product line’s Cost, Revenue and Profit models derived
earlier, verify the following principle from economics: at the value of
“x” (units produced and sold) where Profit “P” is a maximum, marginal
Cost “mC” = marginal Revenue “mR”.

6. Using differential calculus where necessary…
a.  …find the value of the independent variable “x” associated with
maximum average PROFIT “aPmax” for this product line.
b.  …develop the product line’s marginal PROFIT [“mP” or (P)']
expression.
c.  …verify the assertion from econometrics that at the value of “x”
associated with a product line’s “aPmax”, average PROFIT and
marginal PROFIT for this product line are equal.

Extra Credit (optional)

EC1. Corporate headquarters originally set your product line’s PROFIT expectation for the next sales period at \$200,000. Is this PROFIT expectation realistic? Support your answer quantitatively and/or graphically.

EC2. The “most appropriate” demand equation for a particular product is found to be x = 2,000 – 0.625(p). Develop this product’s coefficient of elasticity expression and its Revenue equation R = f(p). Then, assuming there are no severe domain restrictions on price, determine the price where maximum Revenue occurs and the price associated with unit elasticity
(? = –1). What do you observe about the two values?

The full solutions, including the associated excel worksheet are available for purchase. The Answer to part 1 section C question 1 is shown below just as a sample of the answers in the full solutions.

Identify which of the two PROFIT models derived above—1st or 2nd
order—is most appropriate for estimating purposes, according to the
“highest percent variation explained” criterion—a criterion based on
[(R2)] or [(R2)adj]. Based on which of the two models you feel is most
appropriate, would you say that the results for the 1st order or 2nd order
model are most realistic?

For the 1st order model, R² = 0.8623

For the 2nd order model, R² = 0.9742

The 2nd order model appears to be more realistic, because its adjusted coefficient of determination is closer to 1. Note that a visual examination of the best-fit 1st order and 2nd order curves makes it clear that the 2nd order model is a better fit.

Subjects: Mathematics -> Calculus
Topic: Problem set solutions for the class Math X 410 Business Applications of Calculus
Level: College / University
Tags:

### Problem set solutions, Math X 410, Business Applications, Calculus

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Project Details
Subjects: Mathematics -> Calculus
Topic: Problem set solutions for the class Math X 410 Business Applications of Calculus
Level: College / University
Tags:

## Problem set solutions, Math X 410, Business Applications, Calculus

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