CT Assessment FA23 (1)
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School
Pennsylvania State University *
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Course
21
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
4
Uploaded by SuperToad4157
Penn State Altoona
MATH 021
Fall 2023
Critical & Analytical Thinking Assessment
Name:
Score:
Use the given cost and price-demand functions below to answer each of the following ques-
tions. All work must be shown in order to receive full credit.
The cost, in dollars, to produce
x
sweaters is
C
(
x
) = 16
x
+ 128,
x
≥
0 and the price-demand function in
dollars per sweater is
p
(
x
) = 56
−
2
x
, 0
≤
x
≤
28.
1. Explain why the cost function is restricted to
x
≥
0 and the price-demand function is restricted to
0
≤
x
≤
28.
2. Find and interpret
C
(0).
3. Find and interpret
C
(20).
4. Find and interpret
C
(20).
5. Find and interpret
p
(5).
6. Find and simplify
R
(
x
).
7. Find and simplify
P
(
x
).
8. Convert
P
(
x
) to standard (vertex) form. (Hint: Complete the Square.)
9. How many sweaters need to be sold to maximize the profit? What is the maximum profit?
10. What price should be charged per sweater in order to maximize the profit?
11. Find and interpret the break-even points.
12. What prices should be charged per sweater in order to break even?
13. Explain why there are two break-even points.
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14. Sketch the graph of
P
(
x
) for 0
≤
x
≤
20. Be sure to label the maximum and the break-even points.
x
y
−
2
0
4
8
12
16
20
−
130
−
120
−
100
−
80
−
60
−
40
−
20
0
20
40
60
80