X 410 “Business Applications of Calculus Problem Set 1

X 410 “Business Applications of Calculus”
PROBLEM SET 1 [100 points]
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
20
100
130
190
240
300
320
380
430
500

-33986
-31792
-9200
790
21418
37728
54000
58208
65840
65050
50000

1

Section A: 1st Order Model

1. [4] Use Microsoft Excel’s Chart feature to graph a plot of the data, assuming P = (x). Add the most appropriate 1st order “trend line”, the equation of this line, and the equation’s coefficient of determination—its
“[(R2)]”.

Profit [P]
P(x)= 215.51x - 26052
R² = 0.8623

100000

Profit P(x) (dollars)

80000
60000
40000

Profit [P]

20000

Linear (Profit [P])

0
-20000
-40000

0

200

400

600

Units (x)

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. [4] Estimate Profit “P” @ “x” = 0 units and “x” = 70 units.
( )

( )
( )

(

)
(

(

)

)

b. [4] 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.
( )

( )

2

( )

( )

( )
( )

c. [4] 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).

(1) For a 1st order model, the optimum profit will always be a corner solution. Because the profit function is upsloping, the optimum
(maximum) profit is at maximum output that can be sold in a period. x = 510.
The profit at this output is
(