Champernowne's number

He is a very famous number called Champernowne's number: 0,1234567891011121314...

Do you see how it is created ? What makes it so interesting ?

A new operational symbol has been developed in addition to +, - , x, :

This symbol is ~ and is represented by

or a time b divided by a minus b.
Find the value of (2~3)~4

Work

For babysitting Monica charges a flat fee of 3$, plus 5$ per hour. The graph shows how much she earns.

1. Write an equation in slope-intercept form of the line


2. What do you think the slope and y-intercept ?


3. How much money will she make if she babysits 5 hours ?


4. What are the effects if Monica changes her rates so that the equation is y = 6x + 2 ?

A famous female mathematician

Sophie Germain ( 1776 – 1831 ) made major contributions to number theory, acoustics, and elasticity. At age thirteen, Sophie read an account of death of Archimedes at the hands of Roman soldier. She was so moved by this story that she decided to become a mathematician. Sadly, her parents felt that her interest in mathematics was inappropriate, so at night she secretly studied the works of Isaac Newton and the mathematician Leonhard Euler.

A single solution ?

According to The Inquisitive Problem Solver, the only positive integer solution to

A r B r C = C r D r E = E r F r G is

8 r 1 r 9 = 9 r 2 r 4 = 4 r 6 r 3,

if we assume that each variable must be a simple digit.

System of equations

1. Describe the graph of 2x + y = 11 and 2x + y = 2. Determine the number of solutions.
2. Is the system consistent and independent,consistent and dependent or inconsistent?
   9x + 12y = 8 and
3. Solve the system using the linear combination method.
   3x - y - z = 4
   x - 3y + z = - 8
   -3x - 3y + z = - 4
   

Experimenting with the heart

Experimenting with real human hearts isn’t possible, but experimenting with accurate mathematical models of the human heart has led to a new understanding of its complex processes. Mathematics and the computers can replace years of experimentation in laboratories. For example, understanding resulting from mathematics greatly speeds up the design and implementation of artificial valves.

Equations based on Hooke’s Law model the geometry of the heart by representing muscle fibers as close curves of different elasticities. The Navier-Stokers Equations, which describe all fluid flows, model blood flow in and around the heart. The fact that the heart’s shape is constantly changing, however, makes the equations especially hard to solve, and a precise solution to the equation can’t be found. Approximate solutions are generated by computers.