Test 2 Review and Practice Solutions MGF 1106 Test 2 Review and Practice Solutions

You will need to have your own calculator for the test. You may not share calculators or use any type of communication device in place of a calculator. Tests may not be made up for any reason other than a mandatory school – sponsored activity for which you must miss class. If you miss one test for any other reason, your final exam score will be substituted for that test. A second missed test is a zero. No homework bonuses are awarded on a test when the final exam is substituted or you receive a zero on a missed test. It is always possible that some questions may not be exactly like those on the practice test (objectives will remain unchanged). Some of these questions may ask you to analyze or synthesize what we’ve learned but if you prepare by studying your notes and your worksheets (as well as this review), you will be ready to complete those questions as well. Be sure you know the vocabulary listed in the Test 2 Vocabulary Handout. Some vocabulary questions will be on the exam.

Questions on the test will be in a random order and not the order that appears in this review.

Be sure to complete each assignment with a score of at least 80% to receive the 10 point bonus.

The following formulas will be provided. The circumference of a circle is C = πd The area of a circle is A = πr2 The volume of a box is 𝑉=𝑙𝑤ℎ Shape Area Square A = s2 Rectangle A = lw Triangle A = 1 2 𝑏ℎ Parallelogram A = bh Trapezoid A = 1 2 (a + b)h

Suggested Text Problems Exam Topics Objective Sections Suggested Text Problems 1) Construct truth tables 3.3, 3.4 p. 207: 19, 21, 23, 25 2) Determine the truth value of a given statement given specific truth values. p. 207: 29, 31 3) Determine under which conditions a compound statement is true. p. 207: 26 4) Determine if a statement is a tautology, self – contradiction, or neither. 5) Determine if two statements are equivalent. 3.5 p. 207: 35 (a) 6) Write the converse, inverse, and contrapositive of a conditional statement. p. 208: 39, 41 7) Negate conditional statements. 3.6 p. 208: 43, 45 8) Negate compound statements. p. 208: 49, 51 9) Recognize specific forms of valid and invalid arguments. 3.7 p. 208: 59   10) Form a valid conclusion based on specific premises. (see notes) 11) Use an Euler diagram to determine the validity of an argument. 3.8 p. 208: 65, 67 12) Solve problems involving complementary and supplementary angles 10.1 p. 616: 21, 23 13) Solve problems involving angle relationships in triangles. 10.2 p. 680: 15 14) Solve applied problems pertaining to similar triangles. p. 681: 23 15) Solve applied problems involving the Pythagorean Theorem. p. 681: 25 16) Solve applied problems pertaining to perimeter. 10.3 p. 681: 35 17) Use area formulas to compute the areas of plane regions. p. 682: 39, 41 18) Solve applied problems pertaining to area. 10.4 p. 682: 47, 49 19) Solve problems involving volume. 10.5 p. 683: 57 20) Solve applied problems using trigonometry. 10.6 p. 683: 65

Work from simplest → complex to construct the truth table. Important Ideas 1) Construct truth tables Work from simplest → complex to construct the truth table. 2) Determine the truth value of a given statement given specific truth values. Or: True when at least one is true And: Only true if both are true If/Then: False only for True → False If and Only If: True for True ↔ True, False ↔ False 3) Determine under which conditions a compound statement is true. Use a truth table. 4) Determine if a statement is a tautology, self – contradiction, or neither. Tautology – all final table values are true Self – Contradiction – all final table values are false 5) Determine if two statements are equivalent. Final truth values are the same (the right most columns) 6) Write the converse, inverse, and contrapositive of a conditional statement. Converse of p → q: q → p Inverse of p → q: ~p → ~q Contrapositive of p → q : ~q → ~p 7) Negate conditional statements. ~ (p → q) ≡ p ∧~q 8) Negate compound statements. ~ (p ∧ q) ≡ ~p ∨ ~q ~ (p ∨ q) ≡ ~p ∧ ~q 9) Recognize specific forms of valid and invalid arguments. Be sure you know the chart in the section 3.7 notes. 10) Form a valid conclusion based on specific premises. See above. 11) Use an Euler diagram to determine the validity of an argument. For an argument to be invalid, you need only to find one diagram that shows the argument is invalid. 12) Solve problems involving complementary and supplementary angles The sum of the measures of two complementary angles is 90. The sum of the measures of two supplementary angles is 180. 13) Solve problems involving angle relationships in triangles. The sum of the measures of the angles of a triangle is 180. 14) Solve applied problems pertaining to similar triangles. Corresponding sides of similar triangles are proportional. 15) Solve applied problems involving the Pythagorean Theorem. 𝑎 2 + 𝑏 2 = 𝑐 2 16) Solve applied problems pertaining to perimeter. The perimeter of a polygon is the sum of the measures of its sides. Find the perimeter and solve the problem. 17) Use area formulas to compute the areas of plane regions. Carefully compute the area. 18) Solve applied problems pertaining to area. Carefully compute the area and solve the problem. 19) Solve problems involving volume. Carefully compute the volume and solve the problem. 20) Solve applied problems using trigonometry. Sine = opp/hyp. Cosine = adj/hyp Tangent = opp/adj

F F F p q r p q T T T T T T T F T F T F T F T T F F F F F T T F T F Note: In order to work some problems on the test, you must construct truth tables. You should be able to construct “p, q” and “p, q, r” truth tables from scratch. This means that you must make sure you memorize the initial columns: p q r T T T T T F T F T T F F F T T F T F F F T F F F   p q T T T F F T F F

Practice Test Solutions

1) Construct a truth table for the following statements. a) p ∧ (~p ∨ q) p q ~p (~p ∨ q) p ∧ (~p ∨ q) T T F T True T F False F T F F

b) (p↔q)→(p∧r) p q r p↔q p∧r (p↔q)→(p∧r) T T T T True T T F F False T F T T F F F T T F T F F F T F F F

2) Determine the truth value for each statement when p is true, q is true, and r is false. ~p ∨ ~q ~T ∨ ~T F ∨ F False b) (p ∨ r)→ ~(p ∧ q) (T ∨ F)→ ~(T ∧ T) T→ ~T T→ F False c) ~[(p↔q) ∨ (q→~r)] ~[(T↔T) ∨ (T→~F)] ~[T ∨ (T→T)] ~[T ∨ T] ~T False  

The statement is true when both p and q have the same truth values. 3) Consider the statement “If you do a little bit each day then you’ll get by, and if you do not do a little bit each day then you won’t (get by).”   Under what conditions is the statement true? p: You do a little bit each day q: You get by (𝑝→𝑞)∧(~p → ~q) p q 𝑝→𝑞 ~p ~q ~p → ~q (𝑝→𝑞)∧(~p → ~q) T T T F True T F False F T F F The statement is true when both p and q have the same truth values. In other words, the statement is true when p and q are both true and when p and q are both false

4) Consider the statement (p ∧ q) ∧ (~p ∨ ~q) 4) Consider the statement (p ∧ q) ∧ (~p ∨ ~q). Is the statement a tautology, self – contradiction, or neither a tautology nor a self - contradiction? p q 𝑝 ∧ 𝑞 ~p ~q ~p ∨ ~q (p ∧ q) ∧ (~p ∨ ~q) T T T F False T F F T F F The statement is a self – contradiction since all final truth values are false.

Not equivalent since the final truth values do not match up. 5) Consider the statements (p ∨ r)→ ~ q, (~p ∧ ~r )→q. Are the statements equivalent? p q r p ∨ r ~ q (p ∨ r)→ ~ q ~p ~r ~p ∧ ~r (~p ∧ ~r )→q T T T T F T T F T F T T F F F T T F T F F F T F F F Not equivalent since the final truth values do not match up.

6) Write the converse, inverse, and contrapositive of each statement. a) If today is Monday, then I do not have to go to school. Converse: If I do not have to go to school, then today is Monday. Inverse: If today is not Monday, then I have to go to school. Contrapositive: If I do have to go to school, then today is not Monday. b) If all birds fly south for the winter, then some bird-watchers are not happy. Converse: If some bird-watchers are not happy, then all birds fly south for the winter. Inverse: If some birds do not fly south for the winter, then all bird watchers are happy. Contrapositive: If all bird watchers are happy, then some birds did not fly south for the winter.

7) Negate the following statement. If today is Monday, then I do not have to go to school. It is Monday and I do have to go to school.

8) Negate the following statements. a) I practice the piano or I play outside. I don’t practice the piano and I don’t play outside. b) I apply myself and I don’t disappoint my parents. I don’t apply myself or I disappoint my parents.

9) Determine if each argument is valid or invalid. a) If we sing loudly, then the neighbors will hear us. If the neighbors hear us, then the neighbors will smile. ∴ If the neighbors are smiling, then we sung loudly. Invalid: Misuse of Transitive b) If it is Thanksgiving, then I will not skip dessert. I skipped dessert. ∴ It is not Thanksgiving. Valid: Contrapositive. c) He is here or I am working from home. He is here. ∴ I am not working from home. Invalid: Misuse of disjunctive

10) Give a valid conclusion based on the premises. a) If I am a full-time student, I cannot work. If I cannot work, I must budget my money carefully. I am a full time student. Therefore, …I must budget my money carefully. b) If a person is older than nine, then that person must pay the adult price to enter the Magic Kingdom. Becca did not pay the adult price to enter the Magic Kingdom. Therefore, …Becca is not older than nine. c) JoAnn is hungry or Bella is tired. JoAnn is not hungry. Therefore, …Bella is tired.

11) Use an Euler diagram to determine the validity of each argument 11) Use an Euler diagram to determine the validity of each argument. Write “valid” or “invalid”. a) All poets appreciate language. All novelists appreciate language. Therefore, all poets are novelists. Invalid: b) All freshmen live on campus. No people who live on campus can own cars. Therefore, no freshman can own a car. Valid:

𝑥+24+2𝑥=90 3𝑥+24=90 3𝑥=66 𝑥=22 24+2 22 = 24+44=68 68° 𝑎𝑛𝑑 22° 12) The measure of an angle is twenty-four more than twice its complement. Find the measure of each angle. 𝑥+24+2𝑥=90 3𝑥+24=90 3𝑥=66 𝑥=22 24+2 22 = 24+44=68 68° 𝑎𝑛𝑑 22°

13) Find the measure of ∠A in the triangle below 13) Find the measure of ∠A in the triangle below. The triangle is not necessarily drawn to scale. 𝑚∠𝐴+𝑚∠𝐵 +𝑚∠𝐶=180 𝑚∠𝐴+78+90=180 𝑚∠𝐴+168=180 𝑚∠𝐴=12 12 ∘

14) A man who is 6 feet tall is standing 20 feet away from a building 14) A man who is 6 feet tall is standing 20 feet away from a building. If the man’s shadow is ten feet long, how tall is the building?

15) A man is constructing a sail in the shape of a right triangle 15) A man is constructing a sail in the shape of a right triangle. The diagonal side of the sail measures 39 meters. The shorter side of the sail measures 15 meters. How long is the other side of the sail? 𝑎 2 + 15 2 = 39 2 𝑎 2 + 225=1521 𝑎 2 =1296 𝑎= 1296 =36 36 𝑚𝑒𝑡𝑒𝑟𝑠

16) A designer must create a decorative border around a picture frame 16) A designer must create a decorative border around a picture frame. The frame is five feet wide and seven feet long. The decorative border costs $12.75 per yard. How much will the border cost? 𝑃=2 5 + 2 7 = 24 𝑓𝑒𝑒𝑡 24 3 =8 𝑦𝑎𝑟𝑑𝑠 12.75∗8=102 $102

17) Find the area of the figure below. 8 7 𝐴=10·3+7·8=30+56=86 𝑚 2

18) On “Decorator’s Challenge”, Tim was inspired to create a circular patio. The patio has a diameter of 16 feet and Tim learned that his material would cost $5.75 per square foot. What is Tim’s cost to create his patio? Please round your answer to the nearest dollar. You may use either the “π” key on your calculator or 3.14 in calculating your answer. 𝐴= 𝜋 𝑟 2 = 𝜋 8 2 =64𝜋 𝐶𝑜𝑠𝑡=64∙𝜋∙5.75=1156.106097 ≈$1156 𝑜𝑟 𝐶𝑜𝑠𝑡=64∙3.14∙5.75=1155.52 ≈$1156 $1156

19) Find the volume of a cargo container measuring 250 feet by 150 feet by 30 feet. 𝑉=250∙150∙30 =1,125,000 𝑐𝑢𝑏𝑖𝑐 𝑓𝑒𝑒𝑡

20) Work problem 35 on page 667. To find the distance across a lake, a surveyor took the following measurements. How far is it across the lake? tan 40= 𝑎 630 𝑎=630 tan 40=528.63 ≈529 529 yards

Extra Practice Problems MGF 1106 Extra Practice Problems

1) Construct a truth table for the following statement. ~(p ∨~ q)

1) Construct a truth table for the following statement. ~(p ∨~ q) p q ~q p ∨ ~q ~(p ∨~ q) T T F T False T F F T True F F

2) Determine the truth value for the statement when p is false, q is true, and r is false. ~p↔(~q ∧r)  

~False↔(~True ∧ False) 2) Determine the truth value for the statement when p is false, q is true, and r is false. ~p↔(~q ∧r) ~False↔(~True ∧ False) True↔(False ∧ False) True ↔ False False  

p: You’re blushing q: You’re sunburned. 3) Consider the statement “You’re blushing or sunburned, and you’re not sunburned. When is the statement true? p: You’re blushing q: You’re sunburned.

The statement is true when p is true and q is false. 3) Consider the statement “You’re blushing or sunburned, and you’re not sunburned. When is the statement true? p: You’re blushing q: You’re sunburned. (𝑝∨𝑞)∧ ~q p q 𝑝∨𝑞 ~q (𝑝∨𝑞)∧ ~q T T T F False T F True F T F F The statement is true when p is true and q is false.

4) Consider the statement [(𝑝→𝑞)∧~q]→~p Is the statement a tautology, self – contradiction, or neither a tautology nor a self - contradiction?

The statement is a tautology since all final truth values are true. 4) Consider the statement [(𝑝→𝑞)∧~q]→~p Is the statement a tautology, self – contradiction, or neither a tautology nor a self - contradiction? p q 𝑝→𝑞 ~q [(𝑝→𝑞)∧~q ~p [(𝑝→𝑞)∧~q]→~p T T T F True T F F T F F The statement is a tautology since all final truth values are true.

5) Consider the statements ~ p → (q ∨ ~r), (r ∧ ~q )→p Are the statements equivalent?

Equivalent since the final truth values match up. 5) Consider the statements ~ p → (q ∨ ~r), (r ∧ ~q )→p Are the statements equivalent? p q r ~p ~r q ∨ ~r ~ p → (q ∨ ~r) ~q r ∧ ~q (r ∧ ~q )→p T T T F T T T F T F T T F F F T T F T F F F T F F F Equivalent since the final truth values match up.

6) Write the converse, inverse, and contrapositive of the statement. If the review session is successful, then no students fail the test.

6) Write the converse, inverse, and contrapositive of the statement. If the review session is successful, then no students fail the test. Converse: If no students fail the test, then the review session is successful. Inverse: If the review session is not successful, then some students fail the test. Contrapositive: If some students fail the test, then the review session is (was) not successful.

7) Negate the following statement. If there is a tax cut, then all people have extra spending money.

7) Negate the following statement. If there is a tax cut, then all people have extra spending money. There is a tax cut and some people do not have extra spending money.

8) Negate the following statement. They see the show and they do not have tickets.

8) Negate the following statement. They see the show and they do not have tickets. They do not see the show or they have tickets.

9) Determine if the argument is valid or invalid. If the defendants DNA is found at the crime scene, then we can have him stand trial. He is standing trial. ∴ We found evidence of his DNA at the crime scene.

9) Determine if the argument is valid or invalid. If the defendants DNA is found at the crime scene, then we can have him stand trial. He is standing trial. ∴ We found evidence of his DNA at the crime scene. Invalid: Fallacy of the converse

10) Give a valid conclusion based on the premises. If all students get requirements out of the way early, then no students take required courses in their last semester. Some students take required courses in their last semester.

10) Give a valid conclusion based on the premises. If all students get requirements out of the way early, then no students take required courses in their last semester. Some students take required courses in their last semester. Therefore, … some students do not get requirements of their way early.

11) Use an Euler diagram to determine the validity of each argument 11) Use an Euler diagram to determine the validity of each argument. Write “valid” or “invalid”. All people who arrive late cannot perform. All people who cannot perform are ineligible for scholarships. Therefore, all people who arrive late are ineligible for scholarships.

11) Use an Euler diagram to determine the validity of each argument 11) Use an Euler diagram to determine the validity of each argument. Write “valid” or “invalid”. All people who arrive late cannot perform. All people who cannot perform are ineligible for scholarships. Therefore, all people who arrive late are ineligible for scholarships. ineligible Valid Can’t perform late

12) The measure of an angle is three less than two times its supplement. Find the measure of each angle.

𝑥+2𝑥−3=180 3𝑥 −3=180 3𝑥=183 𝑥=61 2 61 −3=122−3=119 61° 𝑎𝑛𝑑 119° 12) The measure of an angle is three less than two times its supplement. Find the measure of each angle. 𝑥+2𝑥−3=180 3𝑥 −3=180 3𝑥=183 𝑥=61 2 61 −3=122−3=119 61° 𝑎𝑛𝑑 119°

13) Find the measure of ∠A in the triangle below 13) Find the measure of ∠A in the triangle below. The triangle is not necessarily drawn to scale. C 96 42 A B

13) Find the measure of ∠A in the triangle below 13) Find the measure of ∠A in the triangle below. The triangle is not necessarily drawn to scale. C 96 42 A B 𝑚∠𝐴+𝑚∠𝐵 +𝑚∠𝐶=180 𝑚∠𝐴+42+96=180 𝑚∠𝐴+138=180 𝑚∠𝐴=42 42 ∘

14) A person standing nine feet from the base of a light pole is five feet tall and casts a shadow that is six feet long. The tip of the shadow is fifteen feet from the base of the lamp post. How tall is the pole?

5 𝑥 = 6 15 6𝑥=75 𝑥=12.5 12.5 𝑓𝑒𝑒𝑡 𝑥 5 9 6 −−−−−−15−−−−−−−−− 14) A person standing nine feet from the base of a light pole is five feet tall and casts a shadow that is six feet long. The tip of the shadow is fifteen feet from the base of the lamp post. How tall is the pole? 5 𝑥 = 6 15 6𝑥=75 𝑥=12.5 12.5 𝑓𝑒𝑒𝑡 𝑥 5 9 6 −−−−−−15−−−−−−−−−

15) Jacob leaned a ladder measuring 15 feet against a vertical wall of his house. The ladder was 11 feet from the base of the wall. How high did the ladder reach? Please round to the nearest tenth of a foot.

15) Jacob leaned a ladder measuring 15 feet against a vertical wall of his house. The ladder was 11 feet from the base of the wall. How high did the ladder reach? Please round to the nearest tenth of a foot. 𝑎 2 + 11 2 = 15 2 𝑎 2 +121=225 𝑎 2 =104 𝑎= 104 =10.2 10.2 𝑓𝑒𝑒𝑡 15 𝑎 11

16) A carpenter is installing a baseboard around a room that has a length of 35 feet and a width of 15 feet. The room has four 3 – foot wide doorways where no baseboard is to be installed. If the cost of the baseboard is $1.50 per foot, what is the cost of installing the baseboard around the room?

16) A carpenter is installing a baseboard around a room that has a length of 35 feet and a width of 15 feet. The room has four 3 – foot wide doorways where no baseboard is to be installed. If the cost of the baseboard is $1.50 per foot, what is the cost of installing the baseboard around the room? 𝑃=2 35 +2 15 =100 𝑆𝑢𝑏𝑡𝑟𝑎𝑐𝑡 𝑜𝑓𝑓 𝑡ℎ𝑒 𝑑𝑜𝑜𝑟𝑠: 100 −4 3 =88 88·1.50=132 $132

17) Find the area of the figure below.

17) Find the area of the figure below. 𝐴=6·3+3·16=18+48=66 𝑓𝑡 2

18) What will it cost to cover a rectangular floor measuring 40 feet by 50 feet with square tiles that measure 2 feet on each side if a package of 10 tiles cost $13?

18) What will it cost to cover a rectangular floor measuring 40 feet by 50 feet with square tiles that measure 2 feet on each side if a package of 10 tiles cost $13? 𝐴=40·50=2000 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑒𝑒𝑡 𝐸𝑎𝑐ℎ 𝑡𝑖𝑙𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑠 2 2 =4 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑒𝑒𝑡 2000 4 =500 𝑡𝑖𝑙𝑒𝑠 500 10 =50 𝑝𝑎𝑐𝑘𝑎𝑔𝑒𝑠 50 𝑝𝑎𝑐𝑘𝑎𝑔𝑒𝑠·$13=$650

19) A train is being loaded with shipping boxes 19) A train is being loaded with shipping boxes. Each box is 8 meters long, 4 meters wide, and 3 meters long. If there are fifty shipping boxes, how much space is needed.

𝑉=8∙4∙3 =96 𝑐𝑢𝑏𝑖𝑐 𝑓𝑒𝑒𝑡 96 ·50=4800 𝑐𝑢𝑏𝑖𝑐 𝑓𝑒𝑒𝑡 19) A train is being loaded with shipping boxes. Each box is 8 meters long, 4 meters wide, and 3 meters long. If there are fifty shipping boxes, how much space is needed? 𝑉=8∙4∙3 =96 𝑐𝑢𝑏𝑖𝑐 𝑓𝑒𝑒𝑡 96 ·50=4800 𝑐𝑢𝑏𝑖𝑐 𝑓𝑒𝑒𝑡

20) A hiker climbs for half a mile (2,640 feet) up a slope whose inclination is 17 degrees. How many feet of altitude, to the nearest foot, did he gain?

20) A hiker climbs for half a mile (2,640 feet) up a slope whose inclination is 17 degrees. How many feet of altitude, to the nearest foot, did he gain? sin 17= 𝑎 2640 𝑎=2640 sin 17 ≈772 772 feet 2,640 a 17 ∘