Problem Solving

Reminder: Quantitative Reasoning

• Amazingly powerful tool to understand the world around us

• Fundamentals:– Ratios– Graphs– Area &Volume– Scaling– Arithmetical statements

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

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

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?

Earth or Sun the Center?

• Aristotle (384–322 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 (310–230 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)

Counter Argument or not?

• Objection to Aristarchus’s model: parallax of stars is not observed (back then)

• Aristarchus argued that this means the stars must be very far away

Measuring the Size of the Earth• Eratosthenes (ca. 276 BC)

– Measures the radius of the earth to about 20%

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

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 Moon’s diameter, then use geometry and same angular size of Earth and Moon to determine Earth-Moon distance

That means we can size it up!

• We can then take distance (384,000 km) and angular size (1/2 degree) to get the Moon’s size

• D = 0.5/360*2π*384,000km = 3,350 km

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

Simple, ingenious idea – hard measurement