The Exponentional Function (5.1)

The Exponentional Function (5.1)

Do the following PROBLEMS from the book
Page 347 #49 thru #58
See the bottom of this page for the MOST IMPORTANT STYLE
PROBLEMS from this section - Plus some EVEN ANSWERS!!!

The formula ABOVE allows us to Calculate
the FINAL VALUE "A" of an
ORIGINAL INVESTMENT "P"
that is given INTEREST "r = Rate in Decimal Form"
for a TIME PERIOD "t = how many years".

The interest is COMPOUNDED "n times per year"

The example below shows:

Starting with $5000
at 2 %  = .02

Compounded Daily  n = 365 times

the Chart below shows the FINAL VALUES

for a number of DAYS after starting the investment:

(Note: the product "nt" can be just the
TOTAL NUMBER of times interest is given.

Especially when "t" is not a WHOLE YEAR.)

Find below the expectations for this section:

***********Find the Final Amount or Find the Initial Amount***************

1)    $7000 - at 4%  - compounded yearly - for years  - results in $8516.570317

2)    $7000 - at 4% - compounded monthly - for 5 years - results in $8546.976158

3)    $7000 - at 4% - compounded daily - for 5 years - results in $8549.72561

4)    $3658  - @ 5% -  compounded daily - for 7 years - becomes $5190.824665

5)    $3658  - @ 5% - compounded monthly - for 7 years - becomes $5187.175879

6)    $3658  - @ 5% - compounded yearly - for 7 years - becomes $5147.17334

7)    $8,187.39725 - at 4% - compounded daily - for 5 years - results in $10,000

8)    $4,493.4866 - at 4% - compounded daily - for 20 years - results in $10,000

******* With Continous Compounding any of the four variables can be found given the other 3   

*****(e = 2.71828182...)*****

9)    $3658  - @ 5% - compounded continuously - for 7 years - becomes $5190.949093

10)     9876 - Decaying continously - @ 6% - for 10 years - becomes 5420.063718

11)    $7000 - at 4% - compounded Continously  - for 5 years - results in $8549.819307

12)    $8187.307531 - at 4% - compounded Continously  - for 5 years - results in $10,000

13)     $4493.289641 - at 4% - compounded Continously - for 20 years - results in $10,000

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FOLDING PAPER

The long standing challenge was that a single piece of paper,

no matter the size, cannot be folded in half more than 7 or 8 times.

 Recently, reports have been made that someone folded a piece of paper in half 13 times.

For the single direction folding case the exact limiting equation is:

where L is the minimum possible length of the material,

t is material thickness, and n is the number of folds possible in one direction.

L and t need to be expressed using the same units.

Alternate Direction Folding has the following limit:

This equation gives the width "W" of a square piece of paper needed to fold a piece of paper "n" times, by folding in alternate directions. The actual equation for alternate folding is more complicated, but this relatively simple formula gives a bound that can not be exceeded and is quite close to the actual limit.