physicsa.com
class code: SPS21 teacher: Mr. Elert
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Physics A: Problem Set 3: Proportional reasoning

recommended reading

High Marks: n/a
Barron's Let's Review: 1.6 Graphing data, 1.7 Direct and inverse proportion
physics.info: Curve fitting, Blank graph paper
Wikipedia: n/a

proportional reasoning

Cartoon figures holding their arms in positions that imitate the various curves. Simplified Chinese text at the top. 函数操 = Function Gymnastics.
  1. A group of similar problems
    1. A six pack of tablet computers cost $250. What would two dozen of these devices cost?
    2. Four landscapers can lay 8 meters of brick sidewalk in 6 hours. How long would it take one landscaper to do the job alone?
    3. The floor of a 3 meter square private office is covered by 100 tiles. How many tiles are needed for a 12 meter square shared office?
    4. The density of water is 1000 kg per cubic meter. What is the density of a cup of water?
    5. A student takes 10 minutes to walk from home to school at her normal pace. If she runs twice as fast as she normally walks, how long does it take her to run from home to school?
    6. Twelfth grade students consume twice as much food during lunch as do first graders. A class of 12 first graders consume 6000 calories. How many calories does a class of 24 twelfth graders consume?
    7. Two students watch an instructional video together. The video lasts 12 minutes. How long does the video last if three students watch it?
    8. A pump at a filling station delivers 12 liters of gasoline for $18. How much gas can be bought from this same pump with $6?
    9. A single prep cook needs 1 hour to peel 120 carrots. How many prep cooks would it take to peel 240 carrots in 20 minutes?
    10. A physics student gets a grade of 80% on the first quiz of the semester and 85% on the second quiz. What grade does the student get on the third quiz?

    1. A six pack of tablet computers cost $250. What would two dozen of these devices cost?

      Answer: $1000

      This problem is an example of a direct relationship. As the number of tablets purchased increases so does the number of dollars spent. Two dozen is four times more than six. Four times more tablets means four times more money.

      24 tablets  = 4× more
      6 tablets

      4 × $250 = $1000

      Here's another approach. When two quantities are directly proportional, their ratio is constant.

      a ∝ b  ⇔ 
      a  = k
      b

      In this case, that ratio is the unit price.

      price  =  constant
      (unit price)
      tablet
      $250  =  x
      6 tablets 24 tablets

      x = $1000

    2. Four landscapers can lay 8 meters of brick sidewalk in 6 hours. How long would it take one landscaper to do the job alone?

      Answer: 24 hours

      This problem is an example of an inverse relationship. As the number of people working on a job decreases, the time it takes to finish the job increases. One person is one fourth of four people. One fourth the workers means four times more work for that one person.

      1 person  = ¼ as many
      4 people

      4 × 6 hours = 24 hours

      Here's another way to think about this. When two quantities are inversely proportional, their product is constant.

      a ∝  1
      b
      		 ⇔  ab = k

      In this case, the constant is the number of person hours.

      number of
      workers      		  ×  hours spent
      working      		  =  constant
      (person hours)

      (4 people)(8 hours) = (1 person)x

      x = 24 hours

    3. The floor of a 3 meter square, private office is covered by 100 tiles. How many tiles are needed for a 12 meter square, shared office?

      Answer: 1600 tiles

      This is a problem where one quantity is proportional to the square of the other. The linear dimensions of the floor are both quadrupling — four times longer and four times wider or sixteen times more area.

      (12 m)2  = 42× more = 16× more
      (3 m)2

      16 × 100 tiles = 1600 tiles

    4. The density of water is 1000 kg per cubic meter. What is the density of a cup of water?

      Answer: 1000 kg/m3

      The density of water is an intensive property, which means it doesn't matter how much of it you have. Having more or less water does not make it more or less dense.

      Mathematically, density is independent of volume or, to say it another way, volume is not a factor that affects the density of water. This is not quite the same thing as saying it's constant. The density of water is affected by temperature, phase, and salinity, so it doesn't have a constant value. Pedantic readers would insist that the problem specify the temperature, phase, and salinity — something like "pure liquid water at 4 °C". The unstated rule of an introductory textbook like this one is…

      Don't overthink problems.

    5. A student takes 10 minutes to walk from home to school at her normal pace. If she runs twice as fast as she normally walks, how long does it take her to run from home to school?

      Answer: 5 minutes

      Most people know intuitively that twice as fast means half the time. This is an example of an inverse relationship. Time is inversely proportional to speed in formal language.

      t ∝  1
      v

      The symbol t for time is obvious. The symbol v for speed comes from the closely related word velocity. This comes from the definition of speed as the rate of change of distance with time.

      v =  ∆s
      ∆t

      The symbol s is used for distance because s is the first letter in the Latin word for distance, spatium. The Greek letter ∆ (delta) is used to indicate that the change in the quantity is what matters.

      Time is the denominator of a fraction on the right side of the equals sign. Speed is the numerator of a fraction on the left side (a fraction where the denominator is implied to be one). Numerator on one side. Denominator on the other. That makes those two quantities inversely proportional.

    6. Twelfth grade students consume twice as much food during lunch as do first graders. A class of 12 first graders consume 6,000 calories. How many calories does a class of 24 twelfth graders consume?

      Answer: 24,000 calories

      There are twice as many twelfth graders and they eat twice as much. Double 6000 calories and double it again.

      2 × 2 × 6,000 calories = 24,000 calories

      Mathematically, we have something like this…

      calories
      consumed      		  =  calories
      per student      		  ×  number of
      students

      Which is essentially this…

      dependent
      variable      		  =  rate  ×  independent
      variable

      Which we could shorthand to…

      y = kx

      In this problem, both the rate (calories per student) and the independent variable (number of students) are doubling. This makes the independent variable (calories consumed) quadruple.

      4y = 2k2x

    7. Two students watch an instructional video together. The video lasts 12 minutes. How long does the video last if three students watch it?

      Answer: 12 minutes

      Don't overthink problems.

    8. A pump at a filling station delivers 12 liters of gasoline for $18. How much gas can be bought from this same pump with $6?

      Answer: 4 liters

      Filling stations sell gasoline like gold — as a commodity. There are no things like connection fees or bracket pricing. The rate is set at the pump and does not vary with amount purchased, customer loyalty, or model of car. One way to solve this problem is by computing the rate (the unit price) set for this pump.

      price  =  constant
      (unit price)
      volume
      $18  =  $1.50/L
      12 L

      Divide the amount of money available by the rate to get the volume dispensed.

      $6  = 4 L
      $1.50/L

      Another way to think about this is that $6 is one third of $18.

      $6  = ⅓ as much
      $18

      Gas is directly proportional to cash.

      gas ∝ cash

      One third less money means one third less gas.

      ⅓ × 12 liters = 4 liters

    9. A single prep cook needs 1 hour to peel 120 carrots. How many prep cooks would it take to peel 240 carrots in 20 minutes?

      Answer: 6 prep cooks

      More workers are needed to get a job done in less time. The number of workers is inversely proportional to the time required to do an amount of work.

      workers ∝  1
      time

      More workers are needed to get more work done. The number of workers is directly proportional to the amount of work to be done.

      workers ∝ work

      Twice as much work…

      240 carrots  = 2× more work
      120 carrots

      in one third the time…

      1 hour  = ⅓ as much time
      20 minutes

      means six times more workers are needed…

      2 × 3 × 1 prep cook = 6 prep cooks

    10. A physics student gets a grade of 80% on the first quiz of the semester and 85% on the second quiz. What grade does the student get on the third quiz?

      Answer: This question has no answer.

  2. The recipe below makes a 20 cm by 20 cm pan of cornbread. You need to scale this recipe up for a commercial kitchen that needs to make a 1 m by 1 m square sheet. How much of each ingredient is needed?
    1. 180 g all purpose flour
    2. 150 g yellow cornmeal
    3. 10 g baking powder
    4. 1 g baking soda
    5. 4 g table salt
    6. 200 g light brown sugar
    7. 90 g frozen corn
    8. 2 large eggs
    9. 110 g unsalted butter
    10. 240 mL buttermilk
    The length and the width of the pan are both five times larger in the commercial kitchen than the home kitchen. (1 meter is 5 times more than 20 centimeters.) That means we'll need 5 × 5 = 25 times more of each ingredient.
    All purpose cornbread
    home commercial ingredient
    180 g 4.5 kg all purpose flour
    150 g 3.75 kg yellow cornmeal
    10 g 250 g baking powder
    1 g 25 g baking soda
    4 g 100 g table salt
    200 g 5 kg light brown sugar
    90 g 2.25 kg frozen corn
    2 50 large eggs
    110 g 2.75 kg unsalted butter
    240 mL 6 L buttermilk

graphing data

  1. simple-pendulums.pdf
    In this experiment, simple pendulums of different lengths were constructed and their periods measured. This data set may look linear, but is it isn't. What effect does length have on the period of a simple pendulum? Plot the data in the space below. Add a best fit curve that shows the relationship.
  2. Nyan Cat
    nyan.pdf
    Nyan Cat: Lost In Space is a game based on the 2011 internet meme of a Pop-Tart-cat hybrid that leaves a flowing rainbow trail behind him. In the game, Nyan Cat runs across the screen, jumping between sausages floating in space at different levels. A player gains points every time Nyan Cat manages to catch a piece of food (candy, cake, donuts, ice cream, or milk) or something else valuable (coins or jewels). If Nyan Cat doesn't land on a sausage he falls to his death and the game is over. At the end of a run, the player's score is displayed along with the distance and duration of the run. The game was played seven times and the results were recorded in the table above right. Plot a graph of distance vs. time and use the line of best fit to determine the speed of Nyan Cat in this game. Follow the guidelines for hand drawn graphs.