Fibonacci videos

These are a couple of videos about Fibonacci numbers that I thought you would like.

 

St Marks is one of the best prep schools in the East.

On Sep 21, 2011 5:57 AM, "Tanton, James" <jamestanton> wrote:
> G’Day All:
>
> Congratulations to Riona R, Luya W, Ryan L, and David E who each win some of Dr. T’s should-be-world-famous double-chocolate coconut fudge brownies for their answers to the MANY ONES puzzler. I’ve written my solutions to it at the end of this e-mail.
>
> The $200 amazon.com puzzler is still out there unsolved and up for grabs! (I’ve repeated it too below)
>
> Now it is time for the next puzzler:
>
> ***THE THOUSANDTH … ****
> 1. A number is called a "repdigit” if all of its digits are the same. (For example, 88888 and 55 and 7 are all repdigits.) What is the 1000th repdigit?
>
> 2. A number if called “repeatful” if it has an even number of digits and its first half of digits is the same as its second half. (For example, 137137 and 7800978009 and 22 are repeatful.) What is the 1000th repeatful number?
>
> 3. A number is a “palindrome” if it reads the same way forwards as it does backwards. (For example, 131 and 187652545256781 and 9 are palindromes.) What is the 1000th palindrome?
> ****
>
> I will give away a really-should-be-galactic-famous double-chocolate coconut fudge brownie to each person who can correctly find each of these three 1000th numbers. (For those away from St. Mark’s, I am afraid to say it will have to be a virtual brownie.)
>
> Submit solutions my way by 5 p.m. Monday September 26th!
>
> Have fun!
>
>
> – Dr. T.
>
> P.s. Here is the outstanding $200 puzzler again:
> *****
> We proved last year that 1 is the only square number in the list 1, 11, 111, 1111, 11111, 111111, 111111,… . Is 1 also the only cube number in the list?
>
> If you think no, then find me a number of the form 1111..11111 that is a perfect cube.
> If you think yes, then prove to me there can be no more.
> ****
>
> p.p.s. Here are my solutions to last week’s puzzler:
>
> ****** THE POWER OF 11111111111111 *****
> Explain in a clear manner why each of the following patterns hold true:
>
> PATTERN 1:
> 11 – 3×3 = 2
> 1111 – 33×33 = 22
> 111111 – 333×333 = 222
> 11111111 – 3333×3333 = 2222
> etc
>
> PATTERN 2:
> 1×9 + 2 = 11
> 12×9 + 3 = 111
> 123×9 + 4 = 1111
> 1234×9 + 5 = 11111
> etc (But what happens when we reach the level of adding 10 and 11 and 12 …?)
>
> PATTERN 3:
> 1×8 + 0 + 1 = 9 x1
> 12×8 + 1 + 2 = 9×11
> 123×8 + 12 + 3 = 9×111
> 1234×8 + 123 + 4 = 9×1111
> 12345×8 + 1234 + 5 = 9×11111
> etc (But what happens when we reach the level of adding 10 and 11 and 12 …?)
> *******
>
> ANSWERS:
> 1. 111111 – 333×333 = 111 (1000 + 1) – 999 x 111
> = 111 (1000 + 1 – 999)
> = 111 (2)
> =222
> This approach clearly generalizes to any size of number.
>
> 2. 12345 x 9 + 6 = 12345 x 10 + 6 – 12345
> = 123456 – 12345
> = 111111
> This approach clearly generalizes to any size of number, though carrying will make the pattern hard to detect for bigger numbers. (So don’t carry!)
>
> 3. 123456 x 8 + 12345 + 6 = 123456 x 9 + 7 – 123456 + 12345 – 1
> = 1111111 – ( 123456 – 12345 + 1) by Pattern 2
> = 1111111 – 111112
> =1000000 – 1
> = 999999
> This approach generalizes to any size of number, though carrying will make the pattern hard to detect for bigger numbers.
>
>
> This message has been scanned for malware by Websense. www.websense.com

In an apple orchard, there are 30 trees per acre, and the average yield
per tree is 400 apples.  For each additional tree planted per acre, the
average yield per tree is reduced by 10 apples.  How many trees per acre
will maximize the crop?

I’ve signed you up for the Mathematica Virtual Conference, which is a free conference the morning on Monday, September 26.  On that day, you will stay home from school to attend the virtual conference.

 

An expression such as y=3x contains two variables. For every
value of x there is a corresponding value of y. The variable x is
called the independent variable and y is called the dependent
variable.

When an increase or decrease in an independent variable leads
to an increase or decrease of the same proportion in the dependent
variable this is termed direct proportion. If y=3x then y is
directly proportional to x, which may be written as y ∝x or y=kx,
where k is called the coefficient of proportionality (in this case,
k being equal to 3).

When an increase in an independent variable leads to a
decrease of the same proportion in the dependent variable (or
vice versa) this is termed inverse proportion. If y is inversely
proportional to x then y ∝1/x or y=k/x. Alternatively, k =xy,
that is, for inverse proportionality the product of the variables is
constant.

Examples of laws involving direct and inverse proportional in
science include:

(i) Hooke’s law, which states that within the elastic limit of
a material, the strain ε produced is directly proportional to
the stress, σ, producing it, i.e. ε ∝ σ or ε =kσ.

(ii) Charles’s law, which states that for a given mass of gas at
constant pressure the volume V is directly proportional to
its thermodynamic temperature T, i.e. V ∝T or V =kT.

(iii) Ohm’s law, which states that the current I flowing through
a fixed resistor is directly proportional to the applied voltage
V, i.e. I ∝V or I =kV .

(iv) Boyle’s law, which states that for a gas at constant temperature,
the volume V of a fixed mass of gas is inversely
proportional to its absolute pressure p, i.e. p ∝ (1/V) or
p=k/V, i.e. pV =k

Question 1: Hooke’s law states that stress σ is directly
proportional to strain ε within the elastic limit of a
material.When, for mild steel, the stress is 25 x 10^6 pascals,
the strain is 0.000125. Determine (a) the coefficient of proportionality
and (b) the value of strain when the stress is
18 x 10^6 pascals.

Question 2. The electrical resistance R of a piece of wire
is inversely proportional to the cross-sectional area A.When
A=5 square milimeters, R=7.02 ohms. Determine (a) the coefficient
of proportionality and (b) the cross-sectional areawhen the
resistance is 4 ohms.

Question 3. Boyle’s law states that at constant temperature,
the volume V of a fixed mass of gas is inversely
proportional to its absolute pressure p. If a gas occupies
a volume of 0.08 cubic meters at a pressure of 1.5 x 10^6 pascals
determine (a) the coefficient of proportionality and (b) the
volume if the pressure is changed to 4 x 10^6 pascals..

Question 4: What is another coefficient of proportionality from something else in your life?

This is Khan Academy talking through an AP Calculus exam problem.  Does this look familiar?  It reminds me of the Eiffel Tower problem.

How long is A4 in the below diagram?

How long is A99?

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