• 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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- Fibonacci number, Golden ratio