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Geometry comes from the Greek geo ("earth") and metron ("measurement"). The Greek mathematician Euclid, often referred to as the "Father of Geometry", was the first to describe a formal system for geometry in his textbook the ‘Elements’ in the 3rd century BC. Euclid’s ‘Elements’ was the main text for teaching mathematics from the time of its publication until the early 20th century!
The Greek philosopher Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore in the 6th century BC. However Indian and Babylonian mathematicians had also known of these theories prior to Thales.
French philosopher, mathematician, and writer René Descartes, 1596–1650 or (latinized: Renatus Cartesius), is credited as the father of analytical geometry, the bridge between algebra and geometry. The Cartesian coordinate system allowing reference to a point in space using a set of numbers was named after him.
Geometry is used in all fields of engineering.
Photo: A European and an Arab practicing geometry in the 15th century. Public domain image from the Wikimedia Commons
Geometry is part of the national engineering unit MEM30012A Apply mathematical techniques in a manufacturing engineering or related environment. More details on unit content from training.gov.au
Required skill: use two dimensional geometry to solve practical problems that involve concepts like perimeter and area, and convert between Cartesian and Polar coordinates, respectively. Solve three dimensional problems involving volume and surface area.
There are no prerequisites required for this module. However, to complete the lessons and quiz below you should have an understanding of arithmetic.
Civil Engineers use geometry for survey calculations to find the location of roads, railways and building sites from known reference points, watch this example video.
Chemical Engineers use geometry to calculate the volume of tanks for the design of chemical manufacturing plants. Watch the following video from the LearnChemE youtube channel for an example on how engineers use geometry.
Electrical Engineers use geometry to calculate the phase relationships of AC electrical current and voltage.
Manufacturing or Industrial Engineers use geometry to plan the layout of manufacturing plants, see this example video.
Mechanical and Structural Engineers use geometry to calculate the mass of machines and structures, see this example calculation video.
Please complete the following lessons.
Angles, degrees, and radians.
Maths is fun lesson on Degrees Angles with a 9 question quiz at the bottom of the page
Khan Academy angle basics series of video lectures - there are 6 short videos in this series and 3 practice sections (starred)
Two dimensional figures perimeter and area.
Maths is fun lesson on Plane Geometry
Khan Academy basic geometry series of video lectures - there are 7 short videos and 2 practice sections (starred)
Three dimensional figures volume and surface area.
Maths is fun lesson on Solid Geometry
Khan Academy solid geometry series of video lectures - there are 5 short videos and 3 practice sections (starred)
Cartesian coordinate system, and the equation of a straight line.
Maths is fun lesson on Cartesian Coordinates with a 10 question quiz at the bottom of the page
Maths is fun lesson on Equation of a Straight Line with a 10 question quiz at the bottom of the page
Khan Academy Descartes and Cartesian coordinates series of video lectures - there are 9 short videos and 5 practice sections (starred)
Khan Academy graphing linear equations series of video lectures - there are 8 short videos and 3 practice sections (starred)
All Khan Academy content is available for free at http://www.khanacademy.org/
Angles, degrees and radians
The Tesla model S electric car has a motor power output of 185 Kilowatts (kW) and a maximum Torque output of 270 Newton metres (Nm). From the equation below, what would be the motor speed in revolutions per minute (rpm) at maximum power and torque?
Given, P= Tω where P = Power (Watts) 1 kW=1000 W, ω= Angular velocity (radians/sec) and T = Torque (Newton metres).
From the power equation, angular velocity rad/sec. To convert to revolutions per minute we apply that 1 revolution=360° =2π radians. Hence, revs/sec. To find the motor speed in revs/minute we apply the fact that there are 60 seconds in each minute leading to ω = 109 x 60 = 6543 revs/min.
Two dimensional figures perimeter and area
A domestic solar power installation requires 2.5 kW of solar capacity made up of ten 250 W panels with dimensions of 1027 mm x 1695 mm (Tindo Solar Karra-250), what would be the minimum roof area to install this system, assuming no gaps between panels?
Area 1 panel = 1027 x 1695= 1740765mm2 or 1.74m2
Total area required = 10 x 1740765=17407650 mm2 = 17.4m2
If a mounting frame was to be built around the perimeter of the panels, what length of steel would be required for the frame?
Assume 2 rows of 5 panels hence total length l=1027 x 5=5135mm, width w = 1695 x 2 = 3390mm hence perimeter = 5135 x 2 + 3390 x 2=17050mm or 17.05m of steel required.
Three dimensional figures volume and surface area
Find the energy required to heat a new swimming pool with dimensions of 25 x 8 x 2 m by 5°C above the ambient temperature. Given the volumetric heat capacity of water at 25°C = 4.179 J/[cm3 K]. (Where K=Kelvin the unit of absolute temperature having the same magnitude as Celsius)
Hence, the heat energy E required in Joules (J) is given by E = 4.179 x V x ΔT where ΔT = 5°C
We need to calculate the water volume V in cm3 V=l x w x d and 1m=100cm hence V=2500 x 800 x 200= 400,000,000 cm3 therefore
If gas costs 4.1c/MJ how much does it cost to use gas to heat our pool by 5°C assuming a 100% efficient heating system?
Cost =8,358 x 4.1=$342.67
Practice exercises:
1. The performance version of the Tesla electric car has a maximum power output of 310 kW and a maximum torque of 600 Nm. Given the power equation above, what would be the motor speed at maximum power and torque in revolutions per minute or rpm?
2. A domestic solar power installation requires 3 kW of solar capacity made up of 250 W panels with dimensions of 1027 mm x 1695 mm (Tindo Solar Karra-250). What would be the minimum roof area required to install this system?
3. To install a swimming pool a hole must be excavated 20 m long x 7 m wide x 2.5 m deep. Calculate the volume of soil that must be removed from the site?
4. Calculate the surface area required for a thermal blanket to cover the water surface of the pool in Q3.
Click here to check your answers
Math.com Free math lessons and homework help.
Interactive Maths Geometry exercises with solutions
SOS Mathematics exercises with solutions
Worked geometry examples and free tutoring at Algebra.com
When you have completed the lessons, verify your knowledge with this multiple choice Geometry Quiz
PLEASE NOTE:
Trigonometry comes from the Greek trigōnon ("triangle") and metron ("measurement"). The Greek mathematicians Euclid and Archimedes in the 3rd century BC were the first to prove trigonometric formulas geometrically.
Throughout history carpenters and masons have known of a quick way to confirm a true ‘right angle’. It is based on the most widely known Pythagorean triple (3, 4, 5) or the ‘Rule of 3-4-5’. From the right angle, run a straight line along one side exactly three units in length, and along the second side exactly four units in length, this will create the third side, or hypotenuse of exactly 5 units in length. The geometric law behind this is the Pythagorean Theorem, e.g., for the triangle to the left
c2 = a2 + b2 or c =
Trigonometry is used in all fields of engineering.
Photo: Parts of a right angle triangle, also known as a 30-60-90 triangle after the internal angles.
Public domain image from the Wikimedia Commons
Trigonometry is part of the national engineering unit MEM30012A Apply mathematical techniques in a manufacturing engineering or related environment. More details on unit content from training.gov.au
Required skill: use trigonometry to solve practical problems, such as the length of the sides and angles in right angle triangles. Use the sine and cosine rules to solve problems of acute and obtuse angled triangles. Use inverse (or reciprocal) trigonometric functions to find angles in triangles using the length of 2 sides. Graph circular (trigonometric) functions and identify their period and amplitude.
There are no prerequisites required for this module. However, to complete the lessons and quiz below you should have an understanding of arithmetic.
Civil Engineers use trigonometry for survey calculations to find the location of roads, railways and building sites from known reference points, watch this video for an example.
Chemical Engineers use trigonometry to calculate the length of piping for the design of chemical manufacturing plants.
Electrical and Electronic Engineers use trigonometric functions to model electrical signals.
Manufacturing or Industrial Engineers use trigonometry to model the cutting forces during machining of metals, see this example video.
Mechanical and Structural Engineers use trigonometry to calculate the direction of loads or forces in machines and structures, see this example calculation video.
Interactive Mathematics
Maths is Fun Trigonometry Index
Khan Academy basic trigonometry series of video lectures there are 4 short videos and 3 practice sections to complete
All Khan Academy content is available for free at http://www.khanacademy.org/
From Mathematics Resources for Education and Industry, registration and login required to access the exercises below.
From the integralmaths.org ‘My home’ home page:
Select;
1. Mathematics Resources for Level 3 Engineering
2. Find a resource by mathematical content
3. Trigonometry and coordinate geometry
Then select the lessons on;
AC Electricity
Dormer window extension to house
Wind vectors ship and helicopter
On completion of the exercises you may wish to review the solutions contained in;
Math.com Free trigonometry lessons and homework help.
Interactive Maths trigonometry exercises with solutions
Maths Tutor exercises for trigonometry with solutions
Free math help in trigonometry
When you have completed the lessons, verify your knowledge with this multiple choice Trigonometry Quiz
PLEASE NOTE:
Knowledge of statistics is like a knowledge of foreign languages or of algebra; it may prove of use at any time under any circumstances. - Sir Arthur Lyon Bowley
Statistics and probability are used in all fields of engineering.
Statistical methods are used to analyse data collected from testing, experiments and from the environment. Probability methods are used to predict the likelihood and risks of failure, likelihood of events such as errors, accidents and cyclones. This analysis allows design changes to keep failures to an acceptable level for the application at hand. Watch Rebecca’s Reliability Engineer career story.
Statistics are also concerned with gaining information about a population from a small sample of that population. For example, in product design engineers may need to know the size of the average, largest or smallest person using a product. This information is usually available in handbooks based on measurements taken from a small group (or sample) of a large group of people.
Photo: Plot of normal distribution curve each coloured band has a width of one standard deviation.
Image licensed under the Creative Commons Attribution 2.5 Generic from Wikipedia
Statistical calculations are part of the national engineering unit MEM30012A Apply mathematical techniques in a manufacturing engineering or related environment. More details on unit content from training.gov.au
Required skill: applying statistics to appropriate and simple engineering situations.
There are no prerequisites required for this module; however, to complete the lessons and quiz below you should have an understanding of algebra.
Civil and Environmental Engineers use the Australian Rainfall and Runoff guide to predict the Average Recurrence Interval (ARI) of floods. This figure provides an estimate of the average period in years between the occurrences of a flood of a given size.
For example, the 10 year ARI event will occur on average once every 10 years: this is equivalent to a 10 year ARI having a 10% probability of occurring in any given year.
See this rainfall intensity calculator for ARI of 1-100 years at this Bureau of Meteorology site.
Manufacturing or Industrial Engineers use probability to predict the total number of defective parts in a production run or a batch from a smaller sample.
Electronic Engineers use probability to predict the reliability of a system from the predicted failure rate of individual electronic components or sub systems.
Mechanical Engineers use probability to predict the life before failure of roller bearings in machinery or your car.
Statistical calculations are used in all fields of engineering to predict things such as the failure rate of products, machines and electronic equipment.
Statistical calculations are also used to determine the average of a sample of measurements for things such as:
Standard deviation is used in Industrial and Manufacturing engineering for the quality control of manufacturing processes, via Statistical process control (SPC), more details from American Society for Quality.
Statistical calculations
Probability
Discrete random variables
Definitions
Mean: arithmetic mean or average: is the sum of a series of values divided by the number of values. More details here.
Median: for a series of numbers, median is the middle number when arranged in ascending order. For an even number of values, i.e., there is no middle number the average of the two middle numbers is used. More details here.
Mode: for a set of numbers, mode is the number that occurs most often. There may be more than one mode. More details here.
To get a better understanding of statistical calculations review these lessons and quiz:
Purplemath lesson on Mean, Median, Mode and Range
and
Khan Academy series on Descriptive Statistics, there are 4 short videos and 4 practice sections
All Khan Academy content is available for free at http://www.khanacademy.org/
Definition
Standard deviation, gives a measure of how closely data is grouped around the mean, more details here.
To get a better understanding of standard deviation review these lessons and quiz:
Maths is Fun Standard Deviation and Variance
and
Khan Academy series of video lectures on Variance and standard deviation there are 12 short videos and 5 practice sections
Interactive graphical tool to plot a histogram from geogebratube.org
All Khan Academy content is available for free at http://www.khanacademy.org/
From Mathematics Resources for Education and Industry, registration and login required to access the exercises below.
From the integralmaths.org ‘My home’ home page:
Select;
1. Mathematics Resources for Level 3 Engineering
2. Find a resource by mathematical content
3. Statistics
Then select the lessons on;
Metal Bar Manufacture,
Wind Power
On completion of the exercises you may wish to review the solutions contained in;
Please refer to the following sites for more information:
Textbook and video presentations, Online Statistics Education: A Multimedia Course of Study, David M. Lane, Rice University.
Engineering & Statistics Handbook more detailed reference from the US National Institute of Standards and Technology NIST.
Open Course Library Intro to Statistics.
LearnEasy MEM30012A Statistics reference page.
Please watch the following video, Statistics vs Calculus, TED Talk by Arthur Benjamin 3min.
When you have completed the lessons, verify your knowledge with this multiple choice Statistics and Probability Quiz
PLEASE NOTE:
Algebra is a type of mathematical shorthand consisting of symbols and signs. These algebraic symbols represent numbers that are not known, ‘unknowns’ or represent unspecified numbers or ‘parameters’. This allows us to state relationships or prove properties no matter what numbers are involved, for example, the quadratic equation has parameters a, b and c, and an unknown variable x.
"Algebra" is derived from the Arabic al-jabr or “restoration”, one of the two operations used by Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī to solve quadratic equations. Al-Khwārizmī was a Persian mathematician, astronomer and geographer during the Abbasid Empire, and a scholar in the House of Wisdom in Baghdad. His work was based on older Indian or Greek sources.
Algebra is used in all fields of engineering.
Photo: Plimpton 322, Babylonian tablet showing a positional number system that greatly aided the solving of rhetorical algebraic equations, the early origin of Algebra. Public domain image from the Wikimedia Commons
Algebra is part of the national engineering unit MEM30012A Apply mathematical techniques in a manufacturing engineering or related environment. More details on unit content from training.gov.au
Required skill: using and applying mathematical formulas, problem solving, calculating.
There are no prerequisites required for this module; however, to complete the lessons and quiz below you should have an understanding of arithmetic, fractions, positive and negative numbers.
Civil Engineers use algebra for survey calculations to find the location of roads, railways and building sites from known reference points, an example is in the following video.
Chemical Engineers use algebra to calculate the mass or volume flow rate of chemicals in the design of chemical manufacturing plants. Watch the following video from the LearnChemE youtube channel for an example flow calculation.
Electronic Engineers use algebra to calculate the properties of electrical circuits such as power consumption.
Manufacturing or Industrial Engineers use algebra to plan the maintenance of production machines.
Mechanical Engineers use algebra to calculate the strength of machine components such as shafts.
Assumed prerequisite knowledge: Arithmetic, fractions, positive and negative numbers. See reference sites on the reference tab if revision is required for prerequisite topics.
Algebraic expression manipulation, substitution, transposition.
To get a better understanding of Algebra review these lessons and quiz:
Polynomials, Quadratic and Cubic equations.
To get a better understanding of Polynomials review these lessons and quiz:
All Khan Academy content is available for free at http://www.khanacademy.org/
From Mathematics Resources for Education and Industry, registration and login required to access the exercises below.
From the integralmaths.org ‘My home’ home page:
Select;
1. Mathematics Resources for Level 3 Engineering
2. Find a resource by mathematical content
3. Algebra
Then select the lessons on;
Bending Metal
Car speedometers
Circuit board manufacture
On completion of the exercises you may wish to review the solutions contained in;
Open Course Library Elementary Algebra.
Paul's Online Math Notes
Interactive Maths exercises with solutions
SOS Mathematics exercises with solutions
Worked examples and free tutoring at Algebra.com
When you have completed the lessons, verify your knowledge with this multiple choice Algebra Quiz
PLEASE NOTE:
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2 May, 2018
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