mae301 - homework F00
MAE 301 Foundations of Secondary School Mathematics
MAE 301 Homework ~~ Fall 2000
For October 30:
- Make up a problem involving the triangle inequality. (p. 47)
Try to be different from the book!
- Explain matrix multiplication (2x2) and show why in general
AB does not equal BA. Is there a multiplicative identity matrix? Do matrices
have multiplicative inverses?
- Let [A,B]=AB-BA. Show that this operation is anti-commutative
- What is the relation between absolute value and distance?
Rephrase the inequalities |x-2| < pi and |x+3| > 1 in terms of distance.
For November 6:
- Larson's Algebra II has problems on p. 143 about
the number of solutions to pairs of linear equations in 2 unknowns.
Analyze 42,43,45,46,58,51,52 BY ROW REDUCTION of the associated
2 x 3 matrix. Explain your work. (I.e. if you use a calculator,
explain your input and output).
- There is a set of problems on 3 equations in 3 unknowns on
pp. 181-182. As above, analyze 14,16,20,22,28,33 BY ROW REDUCTION
of the associated
3 x 4 matrix.
- Make up a system of 3 equations in 3 unknowns where there is
a 2-parameter set of solutions. It is hard not to have this fact
obvious on cursory inspection of the system, but try.
- Make up a system of 4 equations in 4 unknowns where there is
a 2-parameter set of solutions. Disguise the nature of the system
with row operations if necessary.
- Explain why a system of 3 linear equations in 4 unknowns cannot
have a unique solution. Argue by ROW REDUCTION.
- Is it true that a system of 3 linear equations in 4 unknowns
must always have at least one solution? Argue by ROW REDUCTION.
For November 13:
Prepare derivations of the following laws:
- Law of cosines
- Law of sines
- Addition law for cosines
- Addition law for sines.
For November 20:
- Explain how a slide-rule performs multiplication.
- Explain how a piano keyboard is like a logarithmic scale.
If the A below middle C has frequency 440 Hertz (cycles per second),
what will be the frequency of the sound made by a bird which sings
an A four octaves higher? Which is the most convenient base
for logarithms in describing the pitch of a musical sound?
- Explain the Richter scale for expressing the strength
- The intensity of sound is measured in decibels. "A decibel is
an arbitrary unit based of the faintest sound that a man can hear.
The scale is logarithmic, so that an increase of 10db means a
tenfold increase in sound intensity; ..." (from Time magazine).
- An ordinary conversation is 60 decibels. A rock music
cocnert (near the amplifiers) is 120 decibels. How many times louder
is the concert?
- The loudness of the water at the foot of Niagara Falls
is a billion times louder that the least audible sound. What
does that come to in decibels?
- We know what a decibel is; what is a bel?
- When do you think the quoted text was written?
For December 4
- Use your calculator to generate 400 random numbers between 0 and 1.
Make and hand in a histogram of the distribution of these numbers
by tenths (between 0 and .1, between .1 and .2, etc.)
- Use these numbers to simluate 20 repetitions of the experiment
of tossing a coin 20 times (let [0,.5]=Heads, [.5,1]=Tails). Suppose
the outcome of each experiment is p heads and q tails. Make and hand
in a histogram of the distribution of p.
- Find, copy and submit an intelligent applicaiton of probability
theory to the Florida Vote Problem. Explain carefully why you think
it is intelligent.
- Write up an analysis of the Birthday Problem: how many people do you
need in a room before the probability of two of them having the same
birthday rises above .5?
Math Dept SUNY Stony Brook
November 27 2000