ddd

Question 1

can be linked to some interesting known numbers or series of numbers.

Answer 1

Patterns in nature

Question 2

Fibonacci created a problem that concerns the

Answer 2

the birth rate of rabbits.

Question 3

The solution to the problem created by Fibonacci is a sequence of numbers called

Answer 3

Fibonacci sequence.

Question 4

The sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 … is calledc

Answer 4

Fibonacci
sequence.

Question 5

The numbers in the sequence are called

Answer 5

Fibonacci numbers.

Question 6

Fibonacci numbers can be observed in some patterns on

Answer 6

sunflowers.

Question 7

The little
florets on the sunflower head has spirals

Answer 7

(counterclockwise and clockwise).

Question 8

Some
sunflowers have

Answer 8

21 and 34 spirals; some have 55 and 89 or 89 and 144 depending on
the species.

Question 9

Golden ratio (also known as

Answer 9

Divine Proportion)

Question 10

(also known as Divine Proportion)

Answer 10

Golden ratio

Question 11

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”.

Answer 11

Golden ratio (also known as Divine Proportion)

Question 12

what is Golden ratio (also known as Divine Proportion)

Answer 12

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”.

Question 13

The blank is given by the irrational number φ

Answer 13

value of the Golden Ratio

Question 14

The value of the Golden Ratio is given by the irrational number φ

Answer 14

1.618.

Question 15

Starting with f next to each other,

Answer 15

two 1 x 1 squares

Question 16

draw a f on top (or below) of the two 1 x 1 squares

Answer 16

2 x 2 square

Question 17

draw a 2 x 2 square on top (or below)
of the two 1 x 1 squares to produce a

Answer 17

3 x 2 rectangle.

Question 18

Then we draw a f next
to the 3 x 2 rectangle to produce a 5 x 3 rectangle.

Answer 18

3 x 3 square

Question 19

Next we draw a f to
produce an 8 x 5 rectangle.

Answer 19

5 x 5 square

Question 20

Just continue adding squares and you will get sets of
rectangles (also called as

Answer 20

golden rectangles

Question 21

If we draw curves through the diagonal of each
square, we create a spiral-like shape known as

Answer 21

Fibonacci spiral.

Question 22

draw a fibonacci spiral

Question 23

the irrational number e is often referred
to as

Answer 23

Euler’s (pronounced “Oiler”)

Question 24

the irrational number e is often referred
to as Euler’s (pronounced “Oiler”) number after the

Answer 24

Swiss mathematician Leonhard Euler

Question 25

Euler also discovered many of its remarkable properties including its being an

Answer 25

irrational number.

Question 26

The number e is also referred to as

Answer 26

Napier’s constant

Question 27

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.

Answer 27

John Napier

Question 28

However, its discovery is attributed to f when he tried to solve a problem related to continuous compound interests.

Answer 28

Jacob Bernoulli

Question 29

Simple interest is interest paid on the

Answer 29

principal only.

Question 29

is a payment charged for borrowing a money or an income for keeping a money in a bank or making an investment.

Answer 29

Interest

Question 30

There are two ways to compute an interest:

Answer 30

simple and compound.

Question 31

on the other hand, is the addition of interest to the original principal.

Answer 31

Compound interest

Question 32

the interest earned also earns interest.

Answer 32

Compound interest

Question 33

where A =

Answer 33

accumulated balance after a time t

Question 34

P =

Answer 34

principal amount

Question 35

r =

Answer 35

interest rate in decimal

Question 36

t =

Answer 36

time in years

Question 37

e =

Answer 37

2.718 (approximately)

Question 38

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?

Answer 38

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.

Question 39

compound interest formula

Answer 39

A = Pert

Question 40

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?

Answer 40

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

Question 41

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?

Question 42

Different models had been formulated to project population growth and one of these is the

Answer 42

Malthusian growth model or simple exponential growth model.

Question 43

the Malthusian growth model is named after

Answer 43

Thomas Robert Malthus.

Question 44

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.

Answer 44

The Malthusian model

Question 45

where P(t) = t Po = r = t = e =

Answer 45

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)

Question 45

the Malthusian growth model formula

Answer 45

P(t) = Poert

Question 46

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.

Answer 46

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.

Question 47

proposed in 1836 an alternate model that allows for a fact that there are constraints in population growth.

Answer 47

Pierre Verhulst

Question 48

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

Answer 48

logistic growth model

Question 49

LOGISTIC MODEL

Answer 49

P(t) = K /1+Ae−kt

Question 50

where P(t) = t K = k =t A =K−P0/ P0 P0 =

Answer 50

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

Question 51

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.

Question 52

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.

Question 53

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

Answer 53

P(t) = Poe−rt

Question 54

From the previous section we learned that a quantity exhibits exponential growth if it increases continuously according to the model

Answer 54

P(t) = Poe rt

Question 55

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.

Answer 55

exponential growth and exponential decay

Question 56

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?

Question 57

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?