Basics Of Mathematical Ability And Investigating Calculations Answer 29633

Introduction : Basics Of Mathematical Ability And Investigating Calculations 

Answer of question 1.4. Apply the rules of indices for positive, negative and fractional indices

Rules for Positiv? Int?g?r Indic?s:

am x an = am+n For Exampl?: 23 x 24 = 27

(am)n = amn For Exampl?: (23)4 = 212

(ab)m = am*bm For Exampl?: (2x3)4 = 24x34

Rules for N?gativ? Int?g?r Indic?s:

am/an = am-n For Exampl?: 25/23 = 22

(1/a)m = 1/am For Exampl?: (1/2)3 = 1/23 = ?

Rules for Fractional Indic?s:

(am)n/m = an For Exampl?: (81)1/4 = 84/4 = 82

(ab)m/n = (am/n)(bm/n) For Exampl?: (2x3)2/3 = (22/3)(32/3)

Answer of question 1.5. Summarise the base units of the SI system and apply the prefix system

Bas? Units are:

M?t?r (m) - l?ngth

Kilogram (kg) - mass

S?cond (s) - tim?

Amp?r? (A) - ?l?ctric curr?nt

K?lvin (K) - t?mp?ratur?

Mol? (mol) - amount of substanc?

Cand?la (cd) - luminous int?nsity

Pr?fix?s are:

Prefixes of SI ar? add?d b?for? th? unit to multiply it by a pow?r of 10. Common on?s are:

giga (G) - 10^9

m?ga (M) - 10^6

kilo (k) - 10^3

c?nti (c) - 10^-2

milli (m) - 10^-3

micro (μ) - 10^-6

nano (n) - 10^-9

for ?xampl?:

kg = 1000 grams

mm = 0. 001 m?t?rs

μs = 0. 000001 s?conds

Answer of question 2.1. Explain natural and base 10 logarithms

Natural Logarithms (bas? ?)

natural log has a bas? of ? ≈ 2. 71828.

Writt?n as ln(x) or log?(x).

Th? natural log of ? is 1 (ln(?) = 1).

Inv?rs? function of ?x.

Bas? 10 Logarithms

Bas? 10 logs hav? a bas? of 10.

Writt?n as log(x) or log of 10(x).

Th? log of 10 its?lf is 1 (log(10) = 1).

The Inv?rs? function of 10x.

Answer of question 2.2. Apply the rules of logarithms by performing two calculations using the product rule and two using the quotient rule

Rul? Of Product:

log(xy) = log(x) + log(y)

log(12) = log(3 * 4)

= log(3) + log(4) = 0. 477 + 0. 602 = 1. 079

log(42) = log(6 * 7) = log(6) + log(7)

= 0. 778 + 0. 845 = 1. 623

Rul? Of Quoti?nt:

log(x/y) = log(x) - log(y)

log(9/3) = log(9) - log(3) = 0. 954 - 0. 477 = 0. 477

log(28/7) = log(28) - log(7)

= 1. 447 - 0. 845 = 0. 602

Answer of question 2.3. Explain the exponential function and perform two calculations illustrating its significance

Th? dramatic capability is charact?riz?d as y = bx, wh?r? b is a positiv? st?ady call?d th? bas?, and x is th? typ?. A f?w c?ntral issu?s are:

It d?monstrat?s r?markabl? d?v?lopm?nt or rot ov?r th? long haul.

Th? bas? b s?ts th? pac? of d?v?lopm?nt/rot. Normal bas?s ar? ? and 10.

It has an int?r?sting r?v?rs? capability call?d th? logarithm.

Th? following ar? two mod?l ?stimations:

Build r?v?nu? A = P(1 + r/n)nt Wh?r? P is h?ad, r is loan f??, n is numb?r of accumulat? p?riods ?ach y?ar, and t tim? in y?ars.

If P=$1, 000, r=5%, n=12, t=3 y?ars:

A = 1000(1 + 0. 05/12)12*3 = $1, 157. 63

Radioactiv? rot Nt = N0?-λt Wh?r? N0 is starting sum, λ is th? rot consist?nt, and t is tim?.

On th? off chanc? that N0=10 mg, λ=0. 2 y?ars-1, t=1 y?ar:

Nt = 10?-0. 2*1 = 10(0. 819) = 8. 19 mg

So th? r?markabl? d?picts d?v?lopm?nt and rot circumstanc?s r?ally contrast?d with straight mod?ls. It has num?rous logical and mon?tary applications.

Answer of question 2.4. Use graphical and algebraic techniques to calculate half lives

Graphical Proc?dur?:

Plot a r?markabl? rot b?nd with th? sum staying on th? y-pivot and tim? on th? x-hub.

Distinguish th? tim? ?st??m (t1⁄2) wh?n th? sum arriv?s at half of th? first sum (N0/2).

This tim? ?st??m is th? half-lif?.

For instanc?, in th? ?v?nt that w? plot a dramatic b?nd for a 100g ?xampl? with a rot consist?nt of 0. 693/10 y?ars. User can outwardly s?? it com?s to 50g l?ftov?r at 10 y?ars. So th? half-lif? is 10 y?ars.

Arithm?tical Strat?gy:

Utiliz? th? rot condition:

Nt = N0*?^(- λt)

Wh?r? λ is th? rot st?ady.

S?t Nt to half of N0 and addr?ss for t:

N0/2 = N0*?^(- λt1⁄2)

t1⁄2 = ln(2)/λ

For an instanc?, on th? off chanc? that λ is 0. 693/10 y?ars:

t1⁄2 = ln(2)/(0. 693/10) = 10 y?ars

So by utilizing graphical inv?stigation or logarithmic ?stimation, w? can without much of a str?tch d?cid? radioactiv? or r?storativ? half-liv?s from rot constants.

Answer of question 3.1. Use Pythagoras theorem to solve three problems in right angled triangles

Th? l?gs of a right triangl? ar? 4 cm and 3 cm in l?ngth. Comput? th? l?ngth of th? hypot?nus?.

According to Pythagoras' Th?or?m:

a2 + b2 = c2

a = 3 cm

b = 4 cm

c2 = 32 + 42

c2 = 9 + 16

c2 = 25

c = √25 = 5 cm

A right triangl?'s hypot?nus? is 10 inch?s in diam?t?r. On? of th? l?gs ?stimat?s 6 inch?s. What is th? proportion of th? oth?r l?g?

c = 10 inch?s

a = 6 inch?s

b2 = c2 - a2

b2 = 102 - 62

b2 = 100 - 36

b2 = 64

b = √64 = 8 inch?s

A right triangl? has a hypot?nus? of l?ngth 15 m and on? l?g of l?ngth 9 m. What ar? th? proportions of th? int?ns? points insid? th? triangl??

Utilizing g?om?try:

cos(B) = a/c = 9/15 = 0. 6 B = cos1(0. 6) = 53° c = 15 m sin(A) = a/c = 9/15 = 0. 6 A = sin1(0. 6) = 37°

Answer of question 3.2. Use sine, cosine and tangent ratios to solve three problems in right angled

Triangles

In a right triangl? diff?r?nt sid?s ar? known: a = 5 cm, and c = 10 cm. Track down th? missing sid? b.

c is th? hypot?nus?.

Utilizing sin?:

sin(A) = a/c

sin(A) = 5/10 = 0. 5

A = 30°

Utilizing cosin?:

cos(A) = b/c

b = cos(30°) * c = 0. 866 * 10 cm = 8. 66 cm

So b = 8. 66 cm

In a 30-60-90 right triangl? th? hypot?nus? c = 20. Vi?w as sid? a.

sin(A) = sid? a/c

sin(60°) = a/20

a/20 = √3/2

a = 10 * √3 = 17. 32 cm

In a triangl? with sid?s a = 14 and b = 9, track down point C.

Utilizing tang?nt:

tan(C) = a/b

tan(C) = 14/9

C = tan−1(14/9) = 53. 13°

Answer of question 3.3. Solve two problems using two and three dimensions

Two dim?nsional Issu? Track down th? r?gion of a circl? with radious 5 cm.

A = πr2

A = π x (5 cm)2

A = π x 25 cm2

A = 78. 54 cm2

Thr?? dim?nsional Issu? Track down th? volum? of a circl? with sw??p 3 m.

V = (4/3)πr3

V = (4/3)π x (3 m)3

V = 36π m3

V = 113. 10 m3

Th? circl? issu? includ?s two asp?cts - w? work out r?gion utilizing l?ngth (radius).

Th? circl? issu? includ?s thr?? asp?cts - w? work out volum? utilizing l?ngth (radius) and applying th? 3-D ?quation.

Answer of question 3.4. Solve one problem using the sine rule and one problem using the cosine rule

Sin? Rul? Issu?:

Giv?n:

c = 10 cm

B = 60°

a =?

Us? sin? rul?:

a/sin(A) = b/sin(B) = c/sin(C)

H?r?:

A = 90° sinc? point inv?rs? sid? c is a right point.

Sin(90) = 1 Apply th? sin? formula:

Cosin? Rul? Probl?m: a/sin(90°) = c/sin(B) a/1 = 10/sin(60°) a = 10 * (1/2) a = 5 cm

Giv?n:

a = 7 inch?s

b = 10 inch?s

C = 45°

c = ?

Apply th? sin? rul?:

c2 = a2 + b2 - 2abcos(C)

Plug giv?n valu?s:

c2 = 72 + 102 - 2(7)(10)cos(45°)

c2 = 49 + 100 - 100cos(45°)

c2 = 149 - 70

c2 = 79

c = √79 = 8. 9 inch?s

Answer of question AC1.5- SI Units and the Prefix System

Key SI Base Units In This Revision Booklet
Quantity	Unit Name	Unit
Time	second	s
Length	meter	m
Mass	kilogram	kg

Key-Derived SI Units
Quantity	Unit Name	Unit
Volume	cubic meter	m³
Speed	meters per second	m/s
Force	newton	N
Acceleration	meters per second squared	m/s²

Rest of the questions

Some converted values are…

4458 cm = 0.04458 km

0.204 kg = 204,000 μg

0.0045 m = 4,500,000,000 nm

11.4 TW = 11,400 GW

640000 mm² = 0.64 m²

25kmh-1 = 6.944 ms-1

Evaluation:

24* 4-2 = 1

x-2*(x1/3)6= 1

In order from smallest to largest, the four values are

W-2, (W3/W4), W0, W3.

Simplification of (2*b1/2)2* 3b3= 12

Reference list

Journals

• Qushem, U.B., Christopoulos, A. and Laakso, M.J., 2022. Learning Management System Analytics on Arithmetic Fluency Performance: A Skill Development Case in K6 Education. Multimodal Technologies and Interaction, 6(8), p.61.
• Reinsburrow, A.L., 2021. Bridging Math Skills and Math Literacy though Task Design and Implementation. Drexel University.
• Forbringer, L. and Weber, W., 2021. RtI in math: Evidence-based interventions. Routledge.
• Hopkins, S., 12. Special Education Pre-Service Teacher Professional Mathematical Noticing. BOOK OF, p.47.
• Allen, R.D., 2022. An Examination of the Effect of Fine Arts Programs on Math Standardized Test Performance: A Quantitative Non-experimental Study (Doctoral dissertation, Northcentral University).
• Fitzhugh II, G., 2019. A comparison of complex thinking required by the elementary New Jersey student learning standards and past New Jersey curriculum standards. Seton Hall University.