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Algebraic Statements And Scaling

Algebraic Statements And Scaling. Scaling. Often one is interested in how quantities change when an object or a system is enlarged or shortened Different quantities will change by different factors!

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Algebraic Statements And Scaling

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  1. Algebraic Statements And Scaling

  2. Scaling • Often one is interested in how quantities change when an object or a system is enlarged or shortened • Different quantities will change by different factors! • Typical example: how does the circumference, surface, volume of a sphere change when its radius changes?

  3. How does it scale? • Properties of objects scale like the perimeter, the area or the volume • Mass scales like the volume (“more of the same stuff”) • A roof will collect rain water proportional to its surface area

  4. Homework: Newton’s Law of Gravity Note that in order to compute a "factor of change" you can ask: by what factor do I have to multiply the original quantity in order to get the desired quantity? Example: Q: By what factor does the circumference of a circle change, if its diameter is halved? A: It changes by a factor 1/2 = 0.5, i.e. (new circumference) = 0.5 * (original circumference), regardless of the value of the original circumference. • If the mass of the Sun was bigger by a factor 2.7, by what factor would the force of gravity change?    scales linear with mass  same factor • If the mass of the Earth was bigger by a factor 2.2, by what factor would the force of gravity change?  scales linear with mass  same factor • If the distance between the Earth and the Sun was bigger by a factor 1.2, by what factor would the force of gravity change?   falls off like the area  factor 1/ f 2 = 1/1.44 = 0.694

  5. Reminder: Quantitative Reasoning • Amazingly powerful tool to understand the world around us • Fundamentals: • Area &Volume • Scaling • Arithmetical statements • Ratios

  6. From Phrase to Equation • Important skill: translate a relation into an equation, and vice versa • Most people have problems with this arithmetical reasoning

  7. Ratios • Different types of ratios • Fractions: 45/7 = 6.42… • Can subtract 7 from 45 six times, rest 3 • With units: 10 ft / 100ft • Could be a (constant) slope, e.g. for every 10ft in horizontal direction have to go up 1 ft in vertical direction • Inhomogeneous ratios: $2.97/3.8 liters

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