Midterm Review Name: M.8A  determine the number of ways an event may occur using combinations, permutations, and the Fundamental Counting Principle

1. An arrangement in which order matters.

2. An arrangement in which order does not matter.

3. A movie theater sells 3 sizes of popcorn (small, medium, and large) with 3 choices of toppings (no butter, butter, extra butter) and then the last option is salt or no salt.  How many possible ways can a bag of popcorn be purchased?

4. Your state issues license plates consisting of letters and numbers.  There are 26 letters and the letters may be repeated.  There are 10 digits and the digits may be repeated.  How many possible license plates can be issued with two letters followed by three numbers?

5. Two cards are drawn at random from a standard deck of 52 cards, without replacement.  What is the probability of drawing a 7 and a king in that order?

6. Throughout history, many people have contributed to the development of mathematics. These mathematicians include Pythagoras, Euclid, Hypatia, Euler, Einstein, Agnesi, Fibonacci, and Pascal. What is the probability that a mathematician’s name selected at random from those listed will start with either the letter E or the letter A?

7. A new bag of golf tees contains 10 red tees, 10 orange tees, 10 green tees and 10 blue tees.  You empty the tees into your golf bag.  What is the probability of grabbing out two tees of the same color in a row for you and your partner?

M.8B  compare theoretical to empirical probability

8. is the measure of how likely an event is to occur.

9. For equally likely outcomes, the of an event, P(E), is the ratio of the number of favorable outcomes to the total number of outcomes possible.

10. is the most accurate scientific "guess" based on the results of experiments to collect data about an event.

11. The complement of an event is the probability of the event happening. Since the sum of all probabilities in sample space is _______, the probability of an event not happening is P(~E) = .

12. A fair die is tossed.  The results appear in the table at the right. Based on this data, what is the empirical probability of tossing a 4?

What is the theoretical probability of tossing a 4?

13. What is the probability of drawing a jack or a red card from a standard deck of playing cards?

14. A coin is tossed two times, then a month of the year is randomly selected. What is the probability of getting tails each time, and a month that starts with the letter J?

15. A card is drawn from a standard deck, not replaced, and another is drawn. What is the probability of choosing a heart then a spade?

16. Geologists say that the probability of a major earthquake occurring in the San Francisco Bay area in the next 30 years is about 90%.  Is this empirical probability or theoretical probability? Justify your answer.

M.8C  use experiments to determine the reasonableness of a theoretical model such as binomial or geometric

17. A experiment consists of n independent trials in which there are only two outcomes: success and failure

18. Jonathon took a multiple choice test with four choices for each question. If he guessed on the last 5 questions, what is the probability that he got exactly three questions correct?

19. Northern Airlines has a reputation for being on time 86% of the time. Find the probability that a Northern Airlines plane will be on time for at least four of its next five flights.

Remember: Probability = # of outcomes # of outcomes

20. The circle represents a balloon on a board at the local carnival with a radius of 5 cm. The square surrounding the balloon has a length of 10 cm. What is the probability that a dart thrown will hit the balloon?

21. The diameter of the bulls-eye is 4 cm. The radius of the middle circle is 6 cm. The radius of the outer circle is 9 cm. What is the probability that a dart thrown at the board will land anywhere inside the middle circle but not the bulls-eye?

M.6C  use the Pythagorean Theorem and special right-triangle relationships to calculate distances

22. To get from point A to point B you must avoid walking through a pond.  To avoid the pond, you must walk 34 meters south and 41 meters east.  To the nearest meter, how many meters would be saved if it were possible to walk through the pond? 

23. A baseball diamond is a square with sides of 90 feet.  What is the shortest distance, to the nearest hundredth of a foot, which the first baseman has to throw to the third baseman to get the runner out?

24. A guy wire supporting a radio tower is positioned 145 feet up the tower. It forms a 45˚ angle with the ground. About how long is the wire?

25. A skate board ramp must be set up to rise from the ground at 30˚. If the height from the ground to the platform is 8 feet, how far away from the platform must the ramp be set?

M.6D  use trigonometric ratios to calculate distances and angle measures

sin = cos = tan =

26. A ladder 6 feet long leans against a wall and makes an angle of 71º with the ground.  Find to the nearest tenth of a foot how high up the wall the ladder will reach.

27. From a point on the ground 25 feet from the foot of a tree, the angle of elevation of the top of the tree is 32º.  Find to the nearest foot, the height of the tree.

28. At 57” from the base of a building you would need to look up 55 to see the top of the building. What ladder length would the local fire department need to rescue someone from the top of this building in case the stairs were not able to be accessed?

M.6A  use similarity, geometric transformations, symmetry, and perspective drawings to describe mathematical patterns and structure in architecture

29. Engineers and architects carefully create plans and blueprints of structures before they build them.  Modern computers can now be used to render three dimensional plan models.  In effect, in building the actual structures, they must be sure that the actual objects are similar to the corresponding object in the plan.  Equivalently, engineers have to be sure that the actual structure is similar to its three-dimensional model.

List the similarities in the architecture of the Taj Mahal.

Does this structure have symmetry?Justify your answer.

30. In the Mondadori Editore building, in Milan what geometric transformations do you see in the architectural structure?

31. In the condos in San Francisco, California what geometric transformations do you see in the architectural structure?

32. The following engraving by Paul Vredemande de Vries is known as the Opera Mathematica Ou Oeuvres Mathematiques Traictand de Geometrie. What mathematical patterns do you see in the architectural structure through the perspective engraving.

M.6B  use scale factors with two-dimensional and three-dimensional objects to demonstrate proportional and non-proportional changes in surface area and volume as applied to architecture and engineering

33. Suppose you had a nice small house shaped like a box. What would happen if we multiplied the width by 10? How much would the volume change?

What is the scale factor change?

What would happen if we multiplied the width and the length by 10? How much would the volume change?

What is the scale factor change?

What would happen if we multiplied the width and the length and height by 10? How much would the volume change?

What is the scale factor change?

M.7A  use trigonometric ratios and functions available through technology to model periodic behavior in art and music

34. Mathematicians have long been known to be attracted to music, the study of which reveals many mathematical relationships. For example, two notes whose frequencies are in the ratio 1:2 are what we now call an octave apart. Middle C on the piano has a frequency of about 262 cycles per second, while the note with double this frequency is the next C higher, or one octave higher. 

If you start with a note with a frequency of 620 Hz, what frequency is two octaves lower than this frequency?

35. Music is also bound integrally with trigonometry. The simplest musical tone, a pure tone, can be modeled by a simple wave. A pure tone, however, can only be produced by an electronic synthesizer. All natural instruments produce compound tones that are also , but that have more complicated wave profiles

M.7B  use similarity, geometric transformations, symmetry, and perspective drawings to describe mathematical patterns and structure in art and photography

36. The following line drawing is by Abalone da Sea Snail. What mathematical patterns do you see in the one point perspective art?

37. Describe the symmetry and similarity yousee in the following artwork (pottery) from theAcoma Pueblo in New Mexico.

38. In a forensic lab when performing a geometric transformation of rotation or resizing, the photographer must use known control points. Proper use of scales allows this process to be objective and then reproducible, thus providing evidence from the crime scene to catch the suspect. Such as a partial print of a shoe. Why would it be important for proper resizing?

M.7C  use geometric transformations, proportions, and periodic motion to describe mathematical patterns and structure in music

39.

According to the above examples complete the following measures in keeping with the given tempo.

40. A Ferris wheel has a radius of 25 feet. The center of the wheel is 30 feet above the ground, and the wheel rotates at a constant speed of 20 feet per second. The height h, in feet, of a passenger above the ground is given as a function of time t, in seconds, with the mathematical model h = 30 + 25 sin(20t). Complete the following table.

Time(sec) 0 4 8 12 16 20 24 28 32Height(feet)

M.7D  use scale factors with two-dimensional and three-dimensional objects to demonstrate proportional and non-proportional changes in surface area and volume as applied to fields

41. Find the total surface area of the dilated figure B and C by a scale factor of 12. Figure A is an 8 x 10 photograph.