MATH 103 American University of Health Sciences Regular Polyhedron Simple Math Problems

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Math 103 – Spring 20 (Virus)
SUNY Buffalo State College
Instructor: G. Kelley
Final Exam
(60 pts)
[Part 1]
Name_________________________
Class Time_______________
I. Place the letter of the correct choice in the blank which best answers each question that follows; then, transfer
your answers to the provided answer sheet: (2 pts/ans)
______ (1) Which of the following is not a property of a regular polyhedron (Platonic Solid)?
(A)
(B)
(C)
(D)
All sides are congruent (identical) regular polygons.
All vertices have exactly the same number of edges and faces emitting from them.
V − E + F = 2 (Euler’s Characteristic) must be satisfied.
Any regular polygon may be a face, with different polygon faces allowable (like a soccer ball.)
______ (2) Which value does not represent the “Golden Ratio”?
(A)
1 5
2
(B)
5
3
(C) 1.618033989…
(D)
6765
4181
______ (3) Which computation is NOT correct?
(A)
(B)
(C)
(D)
(12)3 ≡ 0 (mod 8)
(5)(8) + 20 ≡ 12 (mod 10)
(13)(6) ≡ 3 (mod 5)
On a mod 7 clock, the time 11 hours past 3 is 0.
______ (4) In a Probability Experiment, a first Event has 12 Outcomes and is followed by a second Event with 10
Outcomes. The Basic Counting Principal tells you that the two events combined have how many
Possible Outcomes?
(A) 22
(B) 2
(C) 120
(D) Can not determine.
______ (5) Which property does not fit the description of the set of natural numbers?
(A)
(B)
(C)
(D)
Can be expressed as the quotient of two integers.
The numbers we count with.
The set is infinite in size
Can be generated by adding one to the number previous to it to get the next number.
______ (6) To the right are listed three consecutive Luca Numbers.
What number should appear next in the series?
(A) 3571
(B) 5778
(C) 9349
843
1364
2207
(D) Cannot be determined
______ (7) For a triangle to be a Golden Triangle, it must be:
(A) a right triangle which has a shorter leg half the length of the hypotenuse.
(B) a right triangle which has a longer leg half the length of the hypotenuse.
(C) a 3, 4, 5 right triangle.
(D) A right triangle which has a longer leg twice the length of the shorter leg.
______ (8) If the largest square is removed from a Golden Rectangle, then:
(A)
(B)
(C)
(D)
the “left over” figure (polygon) is a smaller square.
the “left over” figure (polygon) is a smaller Golden Rectangle.
the “left over” figure (polygon) is a smaller Golden Triangle.
the “left over” figure (polygon) is a smaller trapezoid.
______ (9) Which statement is False?
(A) 5280 feet = 1 mile (B) 500 sheets of paper are 2 inches thick.
(C) 27 ≡ 7(mod 10)
(D) 28 = 16
______(10) If a single very large sheet of paper is folded 8 times, how many individually stacked single
sheets of copy paper should have the same (equal) height?
(A) 1024
(B) 32768
(C) 256
(D) 512
I. (Continued)
[Page 2]
______ (11) Which number that follows is an Integer?
(B) −58.9
(A)
(C) 780.8
(D) −42
______ (12) Which number that follows is a Rational Number?
(A) 29.36366366636666…
(B)
−85.777777777…
(C)
π
______ (13) Which number that follows is an Irrational Number?
(A) 23.03003000300003000003…
(B) 38.0269
(C) −29.363636363636363636…
______ (14) What number “type” or “types” is the number −735 ?
(A) Rational (B) Irrational (C) Rational, Integer, and Real (D) Whole and Integer
— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — -II. Write T in the blank if the statement is true; F if the statement is false. (2 pts/ans)
______ (15) There are only 5 Platonic Solids
______ (16) The “dual” Platonic Solid for the Cube is the Icosahedron.
______ (17) The Tetrahedron’s “dual” Platonic Solid is itself.
______ (18) On a mod 7 clock, the time 16 hours after 4 o’clock is
6 o’clock.
______ (19) The set of Irrational Numbers contains decimals
which may Terminate or Repeat.
______ (20) All the numbers in the following list are Rational
Numbers : -3.252552555 25555… 9.27
-47.32222
8.35353535353535…
______ (21) All the numbers list in Problem (20) are Real Numbers.
______ (22) The set {0, 1, 2, 3, 4, 5, 6, 7, …} correctly displays the set of Natural or Counting Numbers.
______ (23) In a combination Probability Experiment which displays 3 events showing individual probabilities as
detailed below, the probability of the three events combined into one experiment should be 1.0
P(Event 1) = 0.5
P(Event 2) = 0.3
P(Event 2) = 0.2
______ (24) The set {… , -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}, which is the set of integers, may be also “listed” in
the format shown here:
{0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, …}.
______ (25) When you roll a pair of dice simultaneously, there are exactly 12 different outcomes that could
occur.
______ (26) The number 23.93833833383333833333… is both an Irrational number and a Real Number.
______ (27) Rational Numbers cannot be negative.
______ (28) Rational Numbers can be decimal “terminators”, decimal “repeaters”, fractions, whole numbers,
counting or natural numbers, or integers.
______ (29) A Platonic Solid may exist with three hexagons emitting from each vertex.
______ (30) The number 237.53533533353333533333… is both an Irrational Number and a Rational Number.
M
Math 103 – Spring 20 (Virus)
SUNY Buffalo State College
Instructor: G. Kelley
Final Exam
(140 pts)
[Part 2]
Name_________________________
Class Time_______________
[Page 3]
III. Solve the following problems. Show your work:
(31) If you could save 5 cents ($0.05) every 30 seconds, approximately how many years would it take
you to save a million dollars? [Hint—How much could you save in 1 year?] (8 pts.)
(32) Determine whether or not this is a true statement. Show work that verifies your conclusion. (8 pts.)
53  6(9)  8  2(mod 4)
(33) Given the following bank routing number, determine what value the “check digit” should be to
make this routing number conform to national standards. (8 pts.)
2
7
1
0
8
1
5
2
X
(Remember: 7 3 9 7 3 9 7 3 9)
(34) The table below shows numbers that satisfy the Pythagorean Theorem.
x
y
z
3
4
5
5
12
13
7
24
25
9
40
41
[Page 4]
a) Fill in the last row on the table by looking for patterns in the numbers shown. What pattern did you
find? Explain the pattern(s) either in words or by formulas. (4 pts.)
b) Verify that your values for x, y, and z in the last row satisfy the Pythagorean Theorem. (4 pts.)
(35) Using the values s =14 and t = 9, determine the Pythagorean Triple that is generated by evaluating
2st, s2 – t2, and s2 + t2 to arrive at the appropriate values of a, b, and c. Then show that the Triple you
generated satisfies the Rule of Pythagoras, a2 + b2 = c2.
(8 pts.)
The table below displays the first 20 Fibonacci numbers.
1 2 3 4 5 6 7
8
9 10 11 12
n
Fn 1 1 2 3 5 8 13 21 34 55 89 144
[Page 5]
13
14
15
16
17
18
19
20
233
377
610
987
1597
2584
4181
6765
(36) (A) State the rule for generating the Fibonacci numbers. You may write it
symbolically or in words. (2 pts.)
(B) Compute F21
(2 pts.)
(C) Using the table above, compute four examples of the expression Fn 1   Fn1  .
(Use n = 2, 3, 4, 5.)
(8 pts.)
2
2
(D) Describe the pattern you found in Part (C) above by giving a general formula. (3 pts.)
Fn1 2  Fn1 2 
[Page 6]
(37) Write 738 as a sum of nonconsecutive (nonadjacent) Fibonacci numbers. (Refer to Problem 36)
(5 pts.)
For this Problem, place your answers directly on the answer sheet you used for
Problems 1 thru 30 [Part 1].
(38) Match the correct Platonic Solid to its correct name:
______ Octahedron
A.
B.
______ Tetrahedron
______ Icosahedron
C.
D.
______ Dodecahedron
______ Cube
E.
(10 pts)
____________________________________________________________________________________
(39) There were 4 million births in the U.S. last year. Explain why during that year we can be certain
that some of those babies were born during the same minute of time. (5 pts.)
[Page 8]
(42) (A) Can 0.12112111211112 . . . be written as a fraction? Why or why not?
(4 pts.)
(B) Give an example of an irrational number between 97.431 and 97.432
(4 pts.)
(C) Express the rational number 2.34343434… as a fraction.
(4 pts.)
(43) Write 738 as a product of primes. (Find the prime factorization of 738)
(4 pts.)
(44) Given this polyhedron, answer questions (A) and (B).
a) Why is this NOT a regular polyhedron? Why is it not a Platonic Solid?
(4 pts.)
b) Show that this polyhedron satisfies Euler’s Characteristic formula for
the vertices, faces, and edges. (8 pts.)
____________________________________________________________________________
(45) What is the probability of rolling a 5 at least once when rolling a single die 16 times? Find your
answer correct to three decimal places or to the nearest tenth of a percent. (Consider the opposite of the
situation described to determine your answer; or, in other words, first find the probability of never
rolling a 5 in 16 rolls of a single die.)
(8 pts.)
[Page 9]
(46) Informally prove that if you start with a golden rectangle and cut off the largest square possible,
the remaining rectangle is also golden. (7 pts.)
____________________________________________________________________________________
(47) The rectangle WZYZ shown is a Golden Rectangle. The segments WX, RT, and ZT have been
removed from the rectangle and redrawn below exactly congruent to their original lengths in the figure.
Those lengths (measurements) of the segments can be found on the Golden Rectangle itself. Informally
prove that these three segments (WX, RT, and ZT) “taken” from this large Golden Rectangle WXYZ are
indeed Golden Sections. (7 pts.)
[End of Exam]

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