ddd Flashcards
can be linked to some interesting known numbers or series of numbers.
Patterns in nature
Fibonacci created a problem that concerns the
the birth rate of rabbits.
The solution to the problem created by Fibonacci is a sequence of numbers called
Fibonacci sequence.
The sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 … is calledc
Fibonacci
sequence.
The numbers in the sequence are called
Fibonacci numbers.
Fibonacci numbers can be observed in some patterns on
sunflowers.
The little
florets on the sunflower head has spirals
(counterclockwise and clockwise).
Some
sunflowers have
21 and 34 spirals; some have 55 and 89 or 89 and 144 depending on
the species.
Golden ratio (also known as
Divine Proportion)
(also known as Divine Proportion)
Golden ratio
exists when a line is divided into two
parts and the ratio of the longer part “a” to shorter part “b” is equal to the ratio of the
sum “a + b” to “a”.
Golden ratio (also known as Divine Proportion)
what is Golden ratio (also known as Divine Proportion)
exists when a line is divided into two
parts and the ratio of the longer part “a” to shorter part “b” is equal to the ratio of the
sum “a + b” to “a”.
The blank is given by the irrational number φ
value of the Golden Ratio
The value of the Golden Ratio is given by the irrational number φ
1.618.
Starting with f next to each other,
two 1 x 1 squares
draw a f on top (or below) of the two 1 x 1 squares
2 x 2 square
draw a 2 x 2 square on top (or below)
of the two 1 x 1 squares to produce a
3 x 2 rectangle.
Then we draw a f next
to the 3 x 2 rectangle to produce a 5 x 3 rectangle.
3 x 3 square
Next we draw a f to
produce an 8 x 5 rectangle.
5 x 5 square
Just continue adding squares and you will get sets of
rectangles (also called as
golden rectangles
If we draw curves through the diagonal of each
square, we create a spiral-like shape known as
Fibonacci spiral.
draw a fibonacci spiral
the irrational number e is often referred
to as
Euler’s (pronounced “Oiler”)
the irrational number e is often referred
to as Euler’s (pronounced “Oiler”) number after the
Swiss mathematician Leonhard Euler
Euler also discovered many of its remarkable
properties including its being an
irrational number.
The number e is also referred to as
Napier’s constant
The number e is also referred to as Napier’s constant after f who introduced
it earlier in a table of appendix for his work on logarithms.
John Napier
However, its discovery is
attributed to f when he tried to solve a problem
related to continuous compound interests.
Jacob Bernoulli
Simple interest is interest paid on the
principal only.
is a payment charged for borrowing a money or an income for keeping a money
in a bank or making an investment.
Interest
There are two ways to compute an interest:
simple
and compound.
on the other hand, is the addition of interest to the original
principal.
Compound interest
the interest earned also earns interest.
Compound interest
where A =
accumulated balance after a time t
P =
principal amount
r =
interest rate in decimal
t =
time in years
e =
2.718 (approximately)
Mary opened a savings account with ₱10,000.00 initial deposit. If the
account earns 8% interest, compounded continuously, how much would be her money
after 3 years? How much would be her money after 3 years if simple interest is applied?
Solution: It is given that P = 10,000 pesos
r = 0.08
t = 3 years
(a) Substituting these values into (1), we have
A = 10,000e
(0.08)3
A = 12,712.49
Hence, Mary’s money after 3 years would be ₱12,712.49.
compound interest formula
A = Pert
How long will it take for ₱2,000 to double if it is deposited in a bank that
pays 3.5% interest rate compounded continuously?
Solution: It is given that P = 2,000 pesos
r = 0.035
A = 4,000 pesos
Substituting these values into (1), we have
4,000 = 2,000e
0.035t
Divide both sides of the equation by 2,000
4,000
2,000 =
2,000e
0.035t
2,000
2 = e
0.035t
e
0.035t = 2
Take the natural logarithm of both sides
lne
0.035t = ln 2
0.035t = ln 2
Divide both sides of the equation by 0.035
0.035t
0.035 =
ln 2
0.035
t = 19.8 years
Robert received a certain amount from his parents as graduation gift. Instead of spending
it, he opened an account that earns 3.5% interest compounded continuously. After 4
years, his account contains ₱23,005.48. How much did Robert receive from his parents
as graduation gift?
Different models had been formulated to project population growth and one of these is the
Malthusian growth model or simple exponential growth
model.
the Malthusian growth model is named after
Thomas Robert Malthus.
is applied in
obtaining population growth of bacteria and even of humans on the assumption that
resources are unlimited and the population has a continuous birth rate throughout time.
The Malthusian model
where P(t) = t
Po =
r =
t =
e =
where P(t) = the population after time t
Po = the initial population
r = the population growth rate in decimals
t = time
e = 2.718 (approximately)
the Malthusian growth model formula
P(t) = Poert
According to United Nation estimates, the total population in the Philippines
for the year 2018 is 106. 51 million, the 13th largest in the world (Philippines
Population, 2018). Census data shows that the population growth rate is
1.52%. Using the Malthusian model, project the population of the Philippines
5 years after.
Solution: It is given that Po = 106.51 million
r = 0.0152
t = 5 years
Substituting these values into (1),
we have
P(5) = 106.51e
0.0152(5)
= 114.92 million
Hence, there will be approximately 114.92 million people in the Philippines by 2023.
proposed in 1836 an alternate model that allows for a fact that there are
constraints in population growth.
Pierre Verhulst
In reality, the growth rate slows down
due to many factors such as diseases, calamity, limited resources, etc. For this reason,
Pierre Verhulst proposed in 1836 an alternate model that allows for a fact that there are
constraints in population growth. The model is known as
logistic growth model
LOGISTIC MODEL
P(t) =
K
/1+Ae−kt
where P(t) = t
K =
k =t
A =K−P0/
P0
P0 =
where P(t) = the population after time t
K = carrying capacity or limiting value
k = relative growth rate coefficient
A =
K−P0
P0
P0 = the initial population at time t = 0
The population of a certain species of fish is modeled by a logistic growth
model with relative growth rate of k = 0.3 . One hundred fish are initially
introduced into the pond with maximum carrying capacity of 500. Assuming
that fish are not harvested,
(a) estimate the number of fish in the pond after one year;
(b) estimate the time it will take for there to be 350 fish in the pond.
Influenza B virus can be spread by direct transmission such as coughing, sneezing or
spitting. Suppose there are two pupils in a class of 40 children who was infected by the
virus. Assuming none of the children has flu vaccine before, estimate (Let the logistic
growth constant k be equal to 0.6030).
(a) the number of children who will catch the virus after 3 days.
(b) estimate the time it will take for 20 children to catch the virus.
On the other hand, if the quantity decreases continuously at a rate r, r > 0, then we have
an exponential decay and it is modeled by the function
P(t) = Poe−rt
From the previous section we learned that a quantity exhibits exponential growth if it
increases continuously according to the model
P(t) = Poe
rt
Notice that the models for f are of the
same form except for the negative sign in the exponent. Examples of exponential decay
are radioactive decay, radiocarbon dating, drug concentration in the blood stream and
depreciation value.
exponential growth and exponential decay
A certain radioactive element has an annual decay rate of 12%. If there is
a 100-gram sample of the element right now, how many grams will be left in
3 years? What is the half-life of the said radioactive element?
Manny takes 500 mg of ibuprofen to relieve pain from arthritis. Each hour, the amount
of ibuprofen in his system decreases by 25%. How much ibuprofen is left after 4 hours?
