CHAPTER 2 MATH TOOL KIT

Transcription

1 CHAPTER 2 MATH TOOL KIT

2 TYPES OF OBSERVATIONS AND MEASUREMENTS We make QUALITATIVE observations changes in color and physical state. We also make QUANTITATIVE MEASUREMENTS, which involve numbers. Use SI units based on the metric system

3 WHAT IS A MEASUREMENT? a comparison between an unknown quantity and a standard.

4 STANDARDS OF MEASUREMENT When we measure, we use a measuring tool to compare some dimension of an object to a standard. For example, at one time the standard for length was the king s foot. What are some problems with this standard?

5 A standard is an exact quantity that people agree to use to compare measurements. Scenario: You and your friend need to move a desk but you're not sure if it will fit in your trunk. You don't have a ruler or other tool on you. How can you quickly determine if the desk will fit?

6 Historically, body parts were used as measuring devices The height of a horse was measured in hands Foot comes from the average length of a person's foot. A pace is the length of a person's step. *What is the main problem with this kind of system?

7 You are probably familiar with some metric system units. A 2 bottle of soda. A of sugar. A stick. a. gram b. liter c. meter

8 SI MEASUREMENT Le Système international d'unités Among countries with non-metric usage, the U.S. is the only country significantly holding out. The U.S. officially adopted SI in 1866.

9 Pay Attention! On 9/23/99, $125,000,000 Mars Climate Orbiter entered Mars atmosphere 100 km lower than planned and was destroyed by heat. 1 lb = 1 N 1 lb = 4.45 N This is going to be the cautionary tale that will be embedded into introduction to the metric system in elementary school, high school, and college science courses till the end of time.

10 UNITS OF MEASUREMENT 7 SI units base units: length mass time temperature amt. of substance elect. current luminous intensity meter, m kilogram, kg seconds, s Kelvin, K mole, mol ampere, A candela, cd (⁰C)

11 DERIVED UNITS Examples: Density = Combine base units mass (g) volume (cm3) Platinum 21.5 g/cm 3 Speed = meters (m) second (s)

12 Creating units of measure What would happen if the tallest person gave directions in their Vador units to the shortest person? Which is more accurate metric or english system?

13 SOME TOOLS FOR MEASUREMENT

14 MASS the amount of matter in an object Measured in grams and kilograms

15 VOLUME Liter Milliliter Beakers and Graduated cylinders used to measure volume

16 READING VOLUME IN A GRADUATED CYLANDER meniscus The free surface of a liquid called meniscus, assumes a flat, convex or concave shape, depending on the solid and liquid surface. Consider the following diagram

17 cc = cubic centimeter, the same as a ml

18 CALCULATIONS USING TEMPERATURE Generally require temp s in kelvins T (K) = t ( C) Body temp = 37 C = 310 K Liquid nitrogen = -196 C = 77 K

19 METRIC PREFIXES Kilo- means 1000 of that unit 1 kilometer (km) = 1000 meters (m) Centi- means 1/100 of that unit 1 meter (m) = 100 centimeters (cm) 1 dollar = 100 cents Milli- means 1/1000 of that unit 1 Liter (L) = 1000 milliliters (ml)

20 METRIC PREFIXES

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22 LEARNING CHECK A rattlesnake is 2.44 m long. How long is the snake in cm? a) 2440 cm b) 244 cm c) 24.4 cm

23 SOLUTION A rattlesnake is 2.44 m long. How long is the snake in cm? b) 244 cm 2.44 m x 100 cm = 244 cm 1 m

24 CONVERSION FACTORS Fractions in which the numerator and denominator are EQUAL quantities expressed in different units Example: 1 hour = 60 minutes HOW MANY MINUTES ARE IN 2.5 HOURS?

25 SOLUTION Conversion factor 2.5 hr x 60 min = 150 min 1 hr cancel By using dimensional analysis / factor-label method, the UNITS ensure that you have the conversion right side up, and the UNITS are calculated as well as the numbers!

26 SCIENTIFIC NOTATION

27 WHAT IS SCIENTIFIC NOTATION? Scientific notation is a way of expressing really big numbers or really small numbers. For very large and very small numbers, scientific notation is more concise.

28 SCIENTIFIC NOTATION CONSISTS OF TWO PARTS: A number between 1 and 10 A power of 10 N x 10 x

29 TO CHANGE STANDARD FORM TO SCIENTIFIC NOTATION Given: 289,800,000 Use: (number between 1 and 10) Answer: x 10 8 (moved 8 places) Given: Use: 5.67 (moved 4 places) Answer: 5.67 x 10-4

30 Converting to standard notation Given: x 10 6 Answer: 5,093,000 (moved 6 places to the right) Given: x 10-4 Answer: (moved 4 places to the left)

31 LEARNING CHECK Express these numbers in Scientific Notation: 1) 405,789 2) ) 3,012,000,000,000 4) )

32 PRACTICE PROBLEMS worksheet page # 1-14

33 Shallow Lake ave. depth 3 ft. precision, accuracy and significant digits MEASUREMENT UNCERTAINTIES

34 RESULTS MUST BE REPRODUCIBLE no result is perfectly exact every measurement is subject to uncertainty Are they within acceptable limits? can use statistical analysis

35 Need to be sure measurements are as precise as possible while taking into account the accuracy of the instrument being used.

36 BE PRECISE AND ACCURATE

37 SIGNIFICANT FIGURES The numbers reported in a measurement are limited by the measuring tool Significant figures in a measurement includes the known digits plus one estimated digit

38 READING A MEASUREMENT. l I.... I 3....I.... I 4.. cm First digit (known) = 2 2.?? cm Second digit (known) = ? cm Third digit (estimated) between Length reported = 2.65 cm or 2.67 cm or 2.69 cm

39 KNOWN AND ESTIMATED DIGITS In 2.67 cm Known digits 2 and 6 are 100% certain The third digit 7 is estimated (uncertain) In the reported length, all three digits (2.67 cm) are significant including the estimated one

40 Always estimate ONE place past the smallest mark!

41 COUNTING SIGNIFICANT FIGURES RULE 1. All non-zero digits in a measured number are significant. Number of Significant Figures cm ft lb m

42 SANDWICHED ZEROS RULE 2. Zeros between nonzero numbers are significant. Number of Significant Figures 50.8 mm min lb m

43 TRAILING ZEROS RULE 3. All final zeros after a decimal point are significant. Number of Significant Figures 2.50 cm yr 4

44 LEADING ZEROS RULE 4. Leading zeros in decimal numbers are NOT significant. Number of Significant Figures mm oz lb ml

45 All of these contain 3 significant digits 1) 238 2) 105 3) ) ) Can help to represent numbers in scientific notation: X 10-3

46 SIGNIFICANT DIGITS IN CALCULATIONS A calculated answer cannot be more precise than the measuring tool. A calculated answer must match the least precise measurement. Significant figures are needed for final answers from 1) adding or subtracting 2) multiplying or dividing

47 ADDING AND SUBTRACTING The answer has the same number of decimal places as the measurement with the fewest decimal places one decimal place two decimal places answer 26.5 one decimal place

48 MULTIPLYING AND DIVIDING do not round (or add zeros) until the end of the calculation. make sure you have the same number of significant figures as the measurement with the fewest significant digits.

49 TRY IT A X 4.2 = 1) 9 2) 9.2 3) B = 1) ) 62 3) 60 C X = X ) ) 11 3) 0.041

50 ZERO AS A MEASURED NUMBER. l I.... I I.... I 5.. cm What is the length of the line? First digit Second digit Last (estimated) digit is 5.?? cm 5.0? cm 5.00 cm

51 TRY THIS Write a number with 4 values that has one significant and one non-significant zero

52 2.3 Visualizing Data CREATING AND INTERPRETING GRAPHS

53 independent variable (manipulated) The one the experimenter controls directly ( I Vary ) dependent variable (responding) Usually what you re measuring. Its value depends on the IV.

54 LINEAR RELATIONSHIPS The dependent variable varies linearly with the independent variable The two variables are directly proportional There is a linear relationship between the two variables. We can show this relationship in an equation: y = mx + b The slope, m, is the ratio of the vertical change to the horizontal change. m = Δy Δx

55 Speed 50 inferred 45 Design 40 Operating 35 Limit

56 Geometric design is an influence on traffic speeds. Geometry-speed relationship: horizontal curves on rural highways Drivers operate as close to their preferred speeds as possible. Publication No. FHWA-SA , 09/2009

57 Stopping Distances for Dry Pavement/Road 1 How is your driving speed related to your stopping distance? The process of stopping your car has two main components: reaction and braking. Loosely speaking Speed stopping = reaction + braking Thinking Distance 2 Braking Distance Overall Stopping Distance 20 mph 20 feet 20 feet 40 feet 30 mph 30 feet 45 feet 75 feet 40 mph 40 feet 80 feet 120 feet 50 mph 50 feet 125 feet 175 feet 60 mph 60 feet 180 feet 240 feet 70 mph 70 feet 245 feet 315 feet 80 mph 80 feet 320 feet 400 feet

58 At high speeds it takes more time to stop, and during that time you are moving faster.

59 QUADRATIC RELATIONSHIP The relationship is NOT linear There is a quadratic relationship between the two variables. We can show this relationship in an equation: y = ax 2 + bx + c This curve exists when one variable depends on the square of another.

60 Does safe following distance = safe stopping distance? you You are following another car with similar performance to yours, and it suddenly slams on the brakes.

61 In this case, you don t care so much about braking distance, because the other car will have a braking distance also. The main factor that determines whether you will have a rear-end collision is just reaction time. The top part of the figure shows the situation at the moment the blue car slams on its brakes. The bottom part of the figure shows the situation after both cars have come to a stop.

62 Scenario: you are in a truck following a sports car

63 There is a moose in the road. You can t see the obstruction, because the blue car is blocking your view. The blue car swerves around the moose. You are unable to swerve because of oncoming traffic (the orange car).

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65 CONCLUSION: Human reaction time for tasks like breaking is about 1.6 seconds. That means that if you follow the three-second rule (maintaining three seconds between you and the car ahead gives you time and distance to respond to problems in the lane ahead of you), you will be able to handle the special case of car following similar car, with a reasonable margin of safety. However, the three second rule is nowhere near enough room to handle a discovered-threat situation (like the moose) or two different vehicle weights/sizes. It is not enough even at 30 mph, even with no margin of safety. It is less than half what you need at 60 mph, even with no margin.

66 INVERSE RELATIONSHIP The relationship is NOT linear There is an inverse (indirect) relationship between the two variables. We can show this relationship in an equation: y = a or xy = a x This curve exists when one variable depends on the inverse of the other.

67 To find out the demand for a product, economists ask people how many units they would be willing to buy (the quantity) at a specific price. Then, they keep changing the price either up or down and ask them how many units they would be willing to buy at this new price. The data is plotted on a chart called a demand schedule.

68 This inverse relationship between the quantity demanded of a product and its price always holds! In fact, it holds so perfectly and is true so often that it is known as the Law of Demand. The Law of Demand is a rule in economics that states when holding everything else constant the price of a product falls, the quantity demanded of the product will increase; and, when the price of a product rises, the quantity demanded of the product will decrease. Intro. Macroeconomic Analysis and Policy, ECON 104, Penn State

69 Reviewing Concepts, pgs Problems 30 40, 46, pgs CHAPTER 2 REVIEW QUESTIONS