EXERCISES AND PROBLEMS

36 Rationalise and simplify if possible. a)

43 Express each of the following intervals as neighbourhoods of the type E(M, r): a) (–3, 3) b) (2, 4) c) (0, 6) d) (–1, 4) e) (–3, 2) f) (0; 7.5) g) (–5; –2.2) h) (1.2; 4.7)

37 State which of these blood test results are outside the reference ranges: a) () loglog log Mx x 32 –=+ b) () loglog loglog Mx y 13 – =+ + c) () ln ln ln ln Mx yx22 3 – =+ +

44 Express M without logarithms in each case. (Remember that if log M = log k, then M = k).

45 An cuboid measures 81827 ## in cm. Calculate:

38 Calculate by applying the definition of logarithm. log4 163 + log4 2 + log 0.0001 + log

39 If log x = 1.3 and log y = 0.8, calculate: a) The total area. Use radicals to express the exact result. b) The length of the diagonal. Use radicals to express this measurement and approximate to one decimal place. c) Calculate the absolute error and relative error of the above approximation.

40 Transform the expression A logarithms and applying their properties, like in the example:

46 In an isosceles triangle the uneven side measures 53 cm and the height referred to that side is one quarter of it. Calculate the perimeter of the triangle. Use radicals to express the result.

Advanced Problem Solving

47 The diameter of the Milky Way is 105 700 light years, and a light year is 1.461 · 1012 km.

a) What is the diameter of our galaxy in kilometres?

Do the same for the following expressions:

Solve Simple Problems

41 Find out for which values of x can the following roots be calculated:

42 The open interval whose midpoint is M and whose endpoints are M - r and M + r is called a neighbourhood with midpoint M and radius r, and it is represented by N(M, r).

For example: E(0, 2) = (–2, 2) and E(2, 3) = (–1, 5) Express the following neighbourhoods as intervals: a) E(0, 1) b) E(0, 3) c) E(3, 5) d) E(–2; 1.5) e) E(–3; 0.3) f) E(2.1; 3) g) E(–0.2; 5.3) b) How many millenia would it take for a spacecraft to cross it if travelling at 2 000 km/s? c) Given that the diameter of an electron is 4 · 10–15 m, how many electrons would it take to form a line all the way around the Milky Way? (Assume it is a circumference).

48 Calculate the height of a regular tetrahedron with 8 cm edges. Use radicals to express the results.

49 Calculate the volume of a regular octahedron with edges measuring 6 cm. Use radicals to express the results.

50 Prove that the ratio between the areas of these two hexagons is 4/3. (Bear in mind that each hexagon can be divided into 6 equilateral triangles).

51 Write in interval form the numbers that verify the following inequalities: a) | x | < 3 b) | x – 1| ≤ 5 c) | x + 3| < 4 How would you express the numbers that verify the inequalities opposite to the previous ones? | x | ≥ 3 | x – 1| > 5 | x + 3| ≥ 4

UNDERSTAND

52 Calculate as a fraction. 1 1 1 2 1 1 1 + + + a) Substitute 2 with 1 + 1/2 and re-calculate. Repeat this process again. b) Check that the numerators and denominators of the results are terms in the Fibonacci sequence. c) What is the approximate value of the quotients obtained?

53 a) Express the terms of the following sequence as a power: aa a 13 13 13 13 13 13 12;; 3 == == b) Check that the exponents are the sum of the terms of a geometric progression with ratio 1/2. c) What is the approximate value of this sequence?

54 If I deposit a sum of money in a bank at 3% annual interest, how many years will it take for it to triple?

You Can Also Try This

55 Check that it is not possible to use a calculator to obtain 5129 · 463 because the number is too large. Use the properties of powers to express it in scientific notation.

56 Calculate the value of x in these expressions: a) loglog x 2237 =+ b) · 75 63 x =+

57 We have taken a 2 cm square and constructed a rectangle, ABFE.

Is it a golden triangle?

Check that the quotient between the sides is the golden number Φ

58 The equation for the golden number is:

Φ2 – Φ – 1 = 0

Use it to check the following equalities: a) Φ2 = Φ + 1 b) Φ – 1 = 1 U c) Φ3 = 2Φ + 1 d) Φ4 = 3Φ + 2

Remember The Theory Reflect

59 True or false? Explain your answer and give examples.

a) Some irrational numbers are not real.

b) There are an infinite number of irrational numbers between two rational numbers.

c) The inverse of a recurring decimal number can be a terminating decimal.

d) The number 0.83 · 109 is not expressed in scientific notation.

60 If ≥ loglogab 2 –33 , what is the relationship between a and b?

61 Look at this way of representing m on the real number line. Explain it.

62 If x is a number of the interval [–1, 3) and y is a number of the interval (0, 4], explain in which interval you might find x + y And x – y ?

63 Are these equalities true or false? Why? a)