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Algebraic Statements And ScalingPowerPoint Presentation

Algebraic Statements And Scaling

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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?

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

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

Reminder: Quantitative Reasoning

- Amazingly powerful tool to understand the world around us
- Fundamentals:
- Area &Volume
- Scaling
- Arithmetical statements
- Ratios

From Phrase to Equation

- Important skill: translate a relation into an equation, and vice versa
- Most people have problems with this arithmetical reasoning

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

- Fractions: 45/7 = 6.42…

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