•
No two snowflakes are exactly alike, due to the
effect of humility and temperature on the ice
crystal as it forms.
•
Snowflakes are example of
fractals
.
Fractals
are objects whose smaller parts look similar to
the bigger part ( when magnified its smaller
parts look like tiny production of the entire
snowflakes).
•
The repetition of shape at the miniscule level is
called
self-similarity
.

Objects in nature which display self-similarity
Ferns
Mountain
Lightning
Root system of plant
Veins and nerve system of
the human body
Clouds

Honeycomb
Cells of honeycombs in the shape of hexagons allow bees to store the
largest quantity of honey given a limited amount of beeswax.
A honeycomb is the nest of
honeybees.
Bees collect nectar
from flowers and deposit them
in honeycomb made from
beeswax, a substance also
derived from honey?

It is the field of mathematical investigation used
to aid decision making in business and industrial
engineering.
The goal of optimization is to
maximize ( or minimize) the quantity of an output,
while at the same time minimizing the quantity of
an output, while at the same time minimizing the
quantity of resources needed to produce it.
OPTIMIZATION
Question:
Illustrate mathematically that the bees have
instinctively found the best solution , evident in the
hexagonal construction of their hives.

Tiger Stripes
The stripes of tigers and zebras, the
spots on hyena’s and other colorations
and patterns displayed by skins of
animals show a regular pattern which
is easily recognized by people.
Allan Turing , the famous
mathematician credited for breaking
Nazi Enigma code in world war II,
proposed that two substances were
behind this pattern. He called them
morphogens.

As of 2017, it is estimated that the world population is about 7.6
billion. Mathematics can be used to model population growth.
FORMULA:
A = Pe
rt
,
where:
A = size of the population after it grows
P
= initial number of people
r = rate of growth and
t = time
World Population

The exponential growth model A = 30e
0.02t
describes the population
of a city in the Philippines in thousands, t years after 1995.
a.
What was the population of the city in 1995?
b.
What will be the population in 2017?
Solution :
Example 1:

2.
The exponential growth model A = 50e
0.07t
describes the
population of a city in the Philippines in thousands, t years after
1997.
a.
What is the population after 20 years?
b.
What is the population in 2037?
Solution :
Example 2:

A.
Determine what comes next in the given patterns.
1.
A, C, E, G, I, _____________
4.
27, 30, 33, 36, 39, _________
2.
27, 30, 33, 36, 39, _________
5.
15, 10, 14, 10, 13 , 10 , _________
3.
3, 6, 12, 24, 48, 96, _________
B.
Substitute the given values in the formula A = Pe
rt
to find the missing
quantity.
6.
P = 680,000; r = 12% per year; t = 8 years
8.
A = 731,093; P = 525, 600
; r = 3% per year
7.
A = 1,240,000; r = 8% per year; t = 30 years
9.
A = 786, 000; P = 247,000
; t = 17 years
C.
Suppose the population of a certain bacteria in a laboratory sample is
100.
If it doubles in population every 6 hours, what is the growth rate?

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