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Name Date Pd Scientific Methods Worksheet 2: Proportional Reasoning Some problems adapted from Gibbs' Qualitative Problems for Introductory Physics 1. One hundred cm are equivalent to 1 m. How many cm are equivalent to 3 m? Briefly explain how you could convert any number of meters into a number of centimeters. xcm 100cm 3m = 300cm One could set up a proportion as in , and solve for x, but it 3m 1m is simpler (and mathematically equivalent) to multiply the number of meters by the factor: 100cm 100cm , as in 3m 300cm 1m 1m 2. Forty-five cm are equivalent to how many m? Briefly explain how you could convert any number of cm into a number of m. 45cm 100cm 45 cm = 0.45 m Again, one could use a proportion and solve for x, but xm 1m that is a 2-step process wheras one could simply make the conversion by multiplying the 1m 1m number of centimeters by the factor , as in 45cm 0.45m 100cm 100cm 3. One mole of water is equivalent to 18 grams of water. A glass of water has a mass of 200 g. How many moles of water is this? Briefly explain your reasoning. 1mole 11.1moles 18g as the new unit. 200g Multiplying by this factor cancels out the g, leaving moles Use the metric prefixes table to answer the following questions: 4. The radius of the earth is 6378 km. What is the diameter of the earth in meters? 6378km giga mega kilo centi milli micro nano 1000m 6,378,000m 1km 6.378 106 m 5. In an experiment, you find the mass of a cart to be 250 grams. What is the mass of the cart in kilograms? 250g Metric prefixes: = 1 000 000 000 = 1 000 000 = 1 000 = 1 / 100 = 1 / 1000 = 1 / 1 000 000 = 1 / 1 000 000 000 billion million thousand hundredth thousandth millionth billionth 1kg 0.25kg 1000g ©Modeling Instruction - AMTA 2013 1 U2 Scientific - ws 2 v3.1 6. How many megabytes of data can a 4.7 gigabyte DVD store? Rather than try to figure out how many megabytes are in a gigabytes (or vice-versa), convert to the base unit with one factor and to the desired unit with a second factor. (use B for byte) 109 B 1MB 4.7GB 6 4.7 103 MB 1GB 10 B 7. A mile is farther than a kilometer. Consider a fixed distance, like the diameter of the moon. Would the number expressing this distance be larger in miles or in kilometers? Explain. One would need more of a smaller unit than of a larger unit for a fixed distance. Consider an easier example. A yard is 3 ft but 36 in. 8. One US dollar = 0.73 Euros (as of 12/13.) Which is worth more, one dollar or one Euro? How many dollars is one Euro? Since one dollar gets you less than one Euro, the Euro is worth more. 1USD 1Euro $1.37 0.73Euro 9. In 2012, Germans paid 1.65 Euros per liter of gasoline. At the same time, American prices were $3.90 per gallon. a. How much would one gallon of European gas have cost in dollars? b. How much would one liter of American gasoline have cost in Euros? (One US dollar = 0.73 Euros, 1 gallon = 3.78 liters) 3.78L 1.65Euro 1USD 1gal $8.54 1gal 1L 0.73Euro 1L 1gal $3.90 0.73Euro 0.75Euro 3.78L 1gal 1USD 10. A mile is equivalent to 1.6 km. When you are driving at 60 miles per hour, what is your speed in meters per second? Clearly show how you used proportions to arrive at a solution. 60mi 1.6km 1000m 1hr 1min 26.7 m This approach uses unitary factors. s 1hr 1mi 1km 60min 60s 1mi 0.444m , which is a useful factor to keep in mind for hr 1s converting speed in miles/hr to m/s. These factors simplify to ©Modeling Instruction - AMTA 2013 2 U2 Scientific - ws 2 v3.1 11. For each of the following mathematical relations, state what happens to the value of y when the following changes are made. (k is a constant) a. y = kx , x is tripled. If k is a constant, then the relationship becomes y x. Tripling x also triples y. b. y = k/x, x is halved If k is a constant, then the relationship becomes y 1/x. Halving x doubles y. 1 1 If y , then ? y 1 . The (?) must be replaced by 2. x x 2 c. y = k/x2 , x is doubled If k is a constant, then the relationship becomes y 1/x2. Doubling x decreases y by a factor of 4. 1 1 1 If y 2 , then ? y ? y 2 . The (?) must be replaced by 1/4. 2 x 4x 2x d. y = kx2, x is tripled. If k is a constant, then the relationship becomes y x2. Tripling x increases y by a factor of 9. ?y 9x If y x 2 , then ? y 3x 2 2 The (?) must be replaced by a 9. 12. When one variable is directly proportional to another, doubling one variable also doubles the other. If y and x are the variables and a and b are constants, circle the following relationships that are direct proportions. For those that are not direct proportions, explain what kind of proportion does exist between x and y. a. y = 3x b. y = ax + b c. y = x d. y = ax2 e. y = a/x f. y = ax g. y = 1/x h. y = a/x2 (a), (c), (f) are direct proportions. In (b), y and x are linearly related, but not proportional. In (d), y is proportional to the square of x. In (e) and (g), y is inversely proportional to x. In (h) y is inversely proportional to the square of x. ©Modeling Instruction - AMTA 2013 3 U2 Scientific - ws 2 v3.1 13. The diagram shows a number of relationships between x and y. a. Which relationships are linear? Explain. (b), (d), (e) and (f) are linear relationships The slope of these graphs are constant. y b. Which relationships are direct proportions? Explain. (b) and (e) - The lines pass through the origin, so a change in x produces the same size change in y. a b c d e c. Which relationships are inverse proportions? Explain. (a) As x increases, y decreases by a constant factor. f x ©Modeling Instruction - AMTA 2013 4 U2 Scientific - ws 2 v3.1