Problem%20Solving

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Problem Solving
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Reminder Quantitative Reasoning

• Amazingly powerful tool to understand the world
  around us
• Fundamentals
• Ratios
• Graphs
• Area Volume
• Scaling
• Arithmetical statements

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From Phrase to Equation

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

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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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Graphs

• Making a graph
• Create a table with values of the independent
  variable and the function
• Draw the coordinate system on a piece of paper
• Put in (x,y) pairs
• Connect the dots
• Example y 3x - 1

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Simple observations profound Questions

• Just using eyes brain can provoke
  cosmological questions
• Is the Earth the center of the Universe?
• How far away are Sun and Moon?
• How big are they?
• How big is the Earth?
• How heavy is the Earth?

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Earth or Sun the Center?

• Aristotle (384322 BC)
• Argued that the planets move on spheres around
  the Earth (geocentric model)
• Argues that the earth is spherical based on the
  shape of its shadow on the moon during lunar
  eclipses
• Aristarchus (310230 BC)
• Attempts to measure relative distance and sizes
  of sun and moon
• Proposes, nearly 2000 years before Copernicus,
  that all planets orbit the Sun, including the
  Earth (heliocentric model)

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Counter Argument or not?

• Objection to Aristarchuss model parallax of
  stars is not observed (back then)
• Aristarchus argued that this means the stars must
  be very far away

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Measuring the Size of the Earth

• Eratosthenes (ca. 276 BC)
• Measures the radius of the earth to about 20

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Documentation discerns subtle Effects

• Hipparchus (190 BC)
• His star catalog a standard reference for sixteen
  centuries!
• Introduces coordinates for the celestial sphere
• Also discovers precession of the equinoxes

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How far away is the Moon?

• The Greeks used a special configuration of Earth,
  Moon and Sun (link) in a lunar eclipse
• Can measure EF in units of Moons diameter, then
  use geometry and same angular size of Earth and
  Moon to determine Earth-Moon distance

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That means we can size it up!

• We can then take distance (384,000 km) and
  angular size (1/2 degree) to get the Moons size
• D 0.5/3602p384,000km 3,350 km

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How far away is the Sun?

• This is much harder to measure!
• The Greeks came up with a lower limit, showing
  that the Sun is much further away than the Moon
• Consequence it is much bigger than the Moon
• We know from eclipses if the Sun is X times
  bigger, it must be X times farther away

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Simple, ingenious idea hard measurement

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