# fall 2013 – MTH 138 – final exam instructor

October 22, 2018

fall 2013 – MTH 138 – final
exam
instructor: M.Schmidtová
100 minutes, no notes, no books, only calculators are allowed
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NAME: score:
50 /

All calculations must be included,
if only results are written they will not be graded.

FORMULAS

ar.as=ar+s ar_as=ar-s .0/msohtmlclip1/01/clip_image002.gif”>
a-n=1/an (an)m=an.m .0/msohtmlclip1/01/clip_image004.gif”>
(ab)n=an.bn (a/b)n=an/bn .0/msohtmlclip1/01/clip_image006.gif”>

(a+b)(a-b)=a2-b2
(a+b)2=a2+2ab+b2 (a+b)3=a3+3a2b+3ab2
+b3
(a-b)2=a2
-2ab+b2
(a-b)3=a3-3a2b+3ab2+b3
x3+y3=(x+y)(x2-xy+y2)
x3-y3=(x-y)(x2+xy+y2)

.0/msohtmlclip1/01/clip_image008.gif”> a(x-x1)(x-x2)=ax2+bx+c
Vx=
-b/(2a)

COMBINATORICS/Counting

Pk(n)=n! : (n-k)!
P’k(n)= nk
Pn(n)=n!
P=n! : (n1!.n2!. … .
nr!)

.0/msohtmlclip1/01/clip_image010.gif”>

.0/msohtmlclip1/01/clip_image012.gif”>

STRAIGHT LINES

y-y1=m.(x-x1)
y=mx+b
ax+by+c=0

Only formulate both
linear programming problems. Do NOT solve them. Give the objective function and all
constraints. [4pts]

a) A company knows that they have to produce
at least 25000 gears and 14500 axles. They have two machines that they can use
to fill these orders. The first costs \$2000 per hour to operate and can produce
1000 gears and 300 axles per hour. The second costs \$2400 per hour to operate
and can produce 600 gears and 600 axles per hour. How many hours should each
machine operate and what is the minimum cost?
[NOTE: Use letters X
and Y to represent unknown variables]

b) A furniture manufacturer makes two types of
coffee tables. A circular table requires 3h of skilled labor and 6h of
unskilled labor and produces a profit of \$50. A rectangular table requires 2h
of each type of labor and yields a profit of \$30. If 90h of skilled labor and
120h of unskilled labor are available for a week, determine how many of each
type of table should be made in order to maximize the profit. [NOTE: Use letters X and Y to represent unknown variables] [4pts]

Solve the system of
linear inequalities graphically: [9pts]

.0/msohtmlclip1/01/clip_image014.gif”>

NOTE: Number each straight line with
corresponding number of inequality. The final solution set
must be highlighted/cross-hatched. Identify/calculate
all vertices of feasible set. Do not give too small figure, you have the entire
page for it. Use ruler if possible.

Twelve graduate students have applied for three available teaching
assistantships. In how many ways can the assistantships be awarded among
these applicants [2+2+3pts]

a) if no preference is given to any student?

b) if one particular student must be awarded
an assistantship?

c) if the group of applicants includes seven
men and five women and it is stipulated that exactly 2 men and one woman must
be awarded an assistantship?

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A company car that has a seating capacity of five is to be used by
five employees. How many possible seating arrangements are there if:
[2+3pts]

a) all five can drive?

b) only three employees can drive?

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Find number of all 6-digits numbers
that can be formed using digits 1, 2, 3, 4, 5, 6,7 with conditions that
repetitions are not allowed and first and last digit has to be even. [4pts]

Graph 3x+2y-9=0.
[2+1pts]

a) Be sure values of x- and y-intercepts are
displayed in the graph.

b) What is the slope of line displayed in a)?

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