Chapter 6 Further Applications (1)

6

Further Applications (1) Contents 6.1 Golden Section 6.2 More about Exponential and Logarithmic 　 Functions 6.3 Nine-Point Circle

6

Further Applications (1)

6.1 Golden Section A. The Golden Ratio A golden section is a certain length that is divided in such a way that the ratio of the longer part to the whole is the same as the ratio of the shorter part to the longer part.

Fig. 6.2

Consider the rod AB as shown in Fig. 6.2, if we say the point C divides the rod in the golden section, then we have

AC CB = . AB AC

Content

This specific ratio is called the golden ratio.

P.2

6

Further Applications (1)

6.1 Golden Section Definition 6.1: The golden ratio is a raio of the sides of a visually appealing rectangle. This ratio approximat ely equals to 1.618 : 1 or 1 : 0.618. The exact value 5 +1 5 −1 of the golden ratio is : 1 or 1 : . 2 2 Examples:

Content

(a)

The largest pyramid in the world, Horizon of Khufu ( 柯孚王之墓 ), is a right pyramid with height 146 m and a square base of side 230 m. The ratio of its height to the side of its base is 146 : 230 ≈ 1 : 1.58.

•

Another famous pyramid, Horizon of Menkaure ( 高卡王之墓 ) , is also a right pyramid with height 67 m and a square base of side 108 m. The ratio of its height to the side of its base is 67 : 108 ≈ 1 : 1.61.

P.3

6

Further Applications (1)

6.1 Golden Section Consider a line segment PQ with length (1 + x)cm. Fig. 6.5(a)

Divide the line segment into two parts such that PR = 1 cm and RQ = x cm. Fig. 6.5(b)

According to the definition of the golden section, we have PR RQ = . PQ PR Therefore, Content

1 x = 1+ x 1 1 = (1 + x) x 0 = x 2 + x − 1 < By quadratic formula −1+ 5 −1− 5 x= or (rejected) 2 2

P.4

6

Further Applications (1)

6.1 Golden Section B. Applications of the Golden Ratio (i) The Parthenon The Parthenon ( 巴特農神殿 ), which is situated in Athens ( 雅典 ), Greece, is one of the most famous ancient Greek temples.

Content Fig. 6.8

L1 : W1 is close to the golden ratio.

P.5

6

Further Applications (1)

6.1 Golden Section (ii) The Eiffel Tower The tower is 320 m high. The ratio of the portion below and above the second floor (l1 : l2 as shown in Fig. 6.9) is equal to the golden ratio.

Content

Fig. 6.9

P.6

6

Further Applications (1)

6.1 Golden Section C. Fibonacci Sequence The Fibonacci sequence is a special sequence that was discovered by a great Italian mathematician, Leonardo Fibonacci ( 斐波那契 ). This sequence was first derived from the trend of rabbits’ growth. Suppose a newborn pair or rabbits A1 (male) and A2 (female) are put in the wild. 1st month : A1 and A2 are growing. 2nd month : A1 and A2 are mating at the age of one month. Another pair of rabbits B1 (male) and B2 (female) are born at the end of this month.

Content

3rd month : A1 and A2 are mating, another pair of rabbits C1 (male) and C2 (female) are born at the end of this month. B1 and B2 are v growing. If the rabbits never die, and each female rabbits born a new pair of rabbits every month when she is two months old or elder, what happens later?

P.7

6

Further Applications (1)

6.1 Golden Section

Content

Fig. 6.12

P.8

6

Further Applications (1)

6.1 Golden Section Definition 6.2: The Fibonacci sequence is a sequence that satisfies the recurrence formula: Tn = Tn −1 + Tn − 2 for n ≥ 3, where T1 = 1, T2 = 1 and Tn stands for the n th term of the Fibonacci sequence. According to the definition of the Fibonacci sequence, the first ten terms of the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

Content

P.9

6

Further Applications (1)

6.1 Golden Section Consider that seven squares with sides 1 cm, 1 cm, 2 cm, 3 cm, 5 cm, 8 cm, 13 cm respectively. Arrange the squares as in the following diagram:

Fig. 6.13

Content

If we measure the dimensions of the rectangles, each successive rectangle has width and length that are consecutive terms in the Fibonacci sequence Then the ratio of the length to the width of the rectangle will tend to the golden ratio.

P.10

6

Further Applications (1)

6.1 Golden Section D. Applications of the Fibonacci Sequence (a) In Music The piano keyboard of a scale of 13 keys as shown in Fig. 6.14, 8 of them are white in colour, while the other 5 of them are black in colour. The 5 black keys are further split into groups of 3 and 2.

Note that the numbers 1,2,3,5,8,13 are consecutive terms of the Fibonacci sequence. Content

Fig. 6.14

In musical compositions, the climax of songs is often found at roughly the phi point (61.8%) of the song, as opposed to the middle or end of the song.

P.11

6

Further Applications (1)

6.1 Golden Section (b) In Nature Number of petals in a flower is often one of the Fibonacci numbers such as 1, 3, 5, 8, 13 and 21.

Content

P.12

6

Further Applications (1)

6.2 More about Exponential and Logarithmic Functions Applications (a) In Economics Suppose we deposited $P in a savings account and the interest is paid k times a year with annual interest rate r%, then the total amount $A in the account at the end of t years can be calculated by the following formula r   A = P 1 +   100k 

Content

kt

In this case, the earned interest is deposited back in the account and also earns interests in the coming year, so we say that the account is earning compound interest.

P.13

6

Further Applications (1)

6.2 More about Exponential and Logarithmic Functions (b) In Chemistry The concentration of the hydrogen ions is indirectly indicated by the pH scale, or hydrogen ion index. pH Value of a solution

[ ]

If H + denotes the hydrogen ion concentrat ion in mol/dm − 3 , then

[ ]

pH = − log H +

For example, if the concentrat ion of hydrogen ions of a solution is 10 − 8 mol/dm − 3, then its pH value is Content

[ ] pH = − log[10 ] pH = − log H +

−8

= −(−8) log 10 log a b = b log a =8

P.14

6

Further Applications (1)

6.2 More about Exponential and Logarithmic Functions (c) In Social Sciences Some social scientists claimed that human population grows exponentially.

Suppose the population P of a city after n years obeys the exponential function P = 20 000 × (1.08) n , where 20 000 is the present population of the city. Content

From the equation, the population of the city after five years will be approximately 29 000.

P.15

6

Further Applications (1)

6.2 More about Exponential and Logarithmic Functions (d) In Archaeology Scientists have determined the time taken for half of a given radioactive material to decompose. Such time is called the half-life of the material. We can estimate the age of an ancient object by measuring the amount of carbon-14 present in the object. Radioactive Decay Formula The amount A of radioactive material present in an object at a time t after it dies follows the formula: −

Content

t h

A = A0 × 2 , Where A0 is the original amount of the radioactive material and h is its half-life.

P.16

6

Further Applications (1)

6.3 Nine-point Circle Theorem 6.1: In a triangle, the feet of the three altitudes, the mid-points of the three sides and the mid-points of the segments from the three vertices to the orthocentre, all lie on the same circle. Fig. 6.17

Content

P.17