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SLIDE 1
Revision direct proportionality
? A car requires 4.5 l to travel 100 km.
What is its fuel consumption in ?
How far can the car travel on a full tank of 40 l?
? Without stopping, a train travels 250 km in 2 hours. How long will it take to complete a journey of 400 km without stopping?
? One magazine has a mass of 275 g. What is the weight in kilograms of 25 such magazines?
SLIDE 2
? A car requires 4.5 l to travel 100 km. What is its fuel consumption in ? How far can the car travel on a full tank of 40 l?
? Without stopping, a train travels 250 km in 2 hours. How long will it take to complete a journey of 400 km without stopping of 400 km?
? One magazine has a mass of 275 g. What is the mass in kilograms of 25 such magazines?
SLIDE 3
Direct proportionality is a relationship between quantities which are variables.
As one quantity goes up, so does the other quantity go up.
Arrow diagram
The density of aluminium is 2700 kg for each cubic metre.
Variables: ,
In each of the following cases, state the two quantities or variables and express these in an arrow diagram.
? A car requires 4.5 l of petrol to travel 100 km.
? Without stopping, a train travels 250 km in 2 hours.
? One magazine has a mass of 275 g.
SLIDE 4
? A car requires 4.5 l to travel 100 km.
Variables:
,
? Without stopping, a train travels 250 km in 2 hours.
Variables: ,
? One magazine has a mass of 275 g.
Variables: ,
SLIDE 5
Proportionality symbol and constant of proportionality
The relation of direct proportionality is expressed by the proportionality symbol, . When one variable is proportional to another, we may introduce a constant of proportionality.
Sometimes, this constant of proportionality is given a name. For example, when the mass of an object is proportional to its volume, we call the constant of proportionality density.
Example
The density of aluminium is 2700 kg for each cubic metre.
Variables: ,
where
We also say, density is mass per unit volume.
Note
The standard symbol for density is the Greek letter, . This is the Greek equivalent to our r, but you can use a p instead.
SLIDE 6
Express each of the following using the proportionality symbol, . Write each as an equation involving a constant of proportionality and define the constant.
? A car requires 4.5 l to travel 100 km.
? Without stopping, a train travels 250 km in 2 hours.
? One magazine has a mass of 275 g.
SLIDE 7
? A car requires 4.5 l to travel 100 km.
Variables:
,
where , is fuel consumption per kilometre
? Without stopping, a train travels 250 km in 2 hours.
Variables: ,
where , is velocity in metres per second
? One magazine has a mass of 275 g.
Variables: ,
where , is mass of one magazine
SLIDE 8
Indirect proportionality
As one variable goes up, the other variable goes down
Suppose the variables are X and Y, then as an arrow diagram this gives
Example
A farmer employs 15 men to harvest a wheat crop. They take 8 days to complete the work. Assuming each man is working at the same rate, how many days would the harvest take if the farmer had only 10 men?
Using the variables and express the relationship between N and T using arrows.
SLIDE 9
A farmer employs 15 men to harvest a wheat crop. They take 8 days to complete the work. Assuming each man is working at the same rate, how many days would the harvest take if the farmer had only 10 men?
Using the variables and express the relationship between N and T using arrows.
Solution
As the number of men goes up, the time required (number of days needed) goes down.
SLIDE 10
A farmer employs 15 men to harvest a wheat crop. They take 8 days to complete the work.
Assuming each man is working at the same rate, how many man-days of work are required to get in the harvest?
SLIDE 11
A farmer employs 15 men to harvest a wheat crop. They take 8 days to complete the work.
Solution
The harvest requires of work.
SLIDE 12
A farmer employs 15 men to harvest a wheat crop. They take 8 days to complete the work. Assuming each man is working at the same rate, how many days would the harvest take if the farmer had only 10 men?
Given that the work requires 120 man days, how long will it take 10 men to complete the work?
SLIDE 13
A farmer employs 15 men to harvest a wheat crop. They take 8 days to complete the work. Assuming each man is working at the same rate, how many days would the harvest take if the farmer had only 10 men?
Given that the work requires 120 man days, how long will it take 10 men to complete the work?
Solution
SLIDE 14
A shipbuilder requires 120 workers to construct a battleship in 18 months. If the battleship must be completed within 8 months, how many workers will be needed?
? Express the relation between workers (W) and time (T) by an arrow diagram.
? Solve the problem.
SLIDE 15
A shipbuilder requires 120 workers to construct a battleship in 18 months. If the battleship must be completed within 8 months, how many workers will be needed?
Solution
?
? Number of man-months required
Workers required to complete within 8 months
SLIDE 16
6 people are required to clean an office in 6 hours. How many cleaners will be required to clean it in 4 hours?
SLIDE 17
6 people are required to clean an office in 6 hours. How many cleaners will be required to clean it in 4 hours?
Solution
Number of man-hours required
People required to complete within 4 hours is
SLIDE 18
Indirect proportion is also called inverse proportion
Relation of inverse proportionality
The expression X is inversely proportional to Y is written
We can introduce a constant of proportionality
A gas is contained within a cylinder.
The pressure (P) a gas exerts against the sides of a container is inversely proportional to its volume (V).
? Express the relation between pressure and volume of a gas using an arrow diagram. Describe this relationship in words.
? Express the relation between pressure and volume of a gas using the proportionality symbol. Write this as an equation using a constant k.
SLIDE 19
The pressure (P) a gas exerts against the sides of a container is inversely proportional to its volume (V).
? Express the relation between pressure and volume of a gas using an arrow diagram. Describe this relationship in words.
As the volume of the gas goes up, the pressure it exerts goes down.
There are many equivalents
? Express the relation between pressure and volume of a gas using the proportionality symbol. Write this as an equation using a constant k.
SLIDE 20
Units of pressure
Pressure is measured in Pascals, with symbol Pa. The units of pressure are usually written in kPa (kilo-Pascals), which are multiples of 1000 Pascals.
The pressure that the air exerts at sea level is on average 100,000 Pa, or 100 kPa, where the k represents 1000. This is also written Pa.
There are many different ways in which pressure is measured; these include mm of mercury. The pressure of the atmosphere is also known as 1 atm.
SLIDE 21
When a gas is contained in a cylinder, the pressure it exerts against the side of the cylinder is inversely proportional to its volume.
? Make k the subject of this equation
The pressure and volume of a gas at one time are and respectively. Later the pressure and volume of the same gas are and .
? Make k the subject of both and
? Eliminate k by equating both expressions to find the relationship between and .
SLIDE 22
?
The pressure and volume of a gas at one time are and respectively. Later the pressure and volume of the same gas are and .
?
? Since k is equal to both we can eliminate k and equate
This equation is known as Boyles Law
SLIDE 23
Boyles Law
A gas in a syringe occupies a volume . Its pressure is 50 kPa. What will its pressure be if the plunger of the syringe is pushed in so that the volume occupied by the gas is reduced to .
SLIDE 24
A gas in a syringe occupies a volume . Its pressure is 50 kPa. What will its pressure be if the plunger of the syringe is pushed in so that the volume occupied by the gas is reduced to
Solution
We have
We substitute into Boyles Law
SLIDE 25
When two objects collide, their combined mass is inversely proportional to their combined velocity .
? Write this relationship using the proportionality symbol.
? Eliminate the proportionality symbol by introducing a constant .
SLIDE 26
When two objects collide, their combined mass is inversely proportional to their combined velocity .
? Write this relationship using the proportionality symbol.
? Eliminate the proportionality symbol by introducing a constant .
SLIDE 27
When two objects collide, their combined mass is inversely proportional to their combined velocity . .
? Make the constant, p, the subject of this equation
The velocity and mass of a moving object at one time are and . This object collides with another, and later, their combined mass and velocity are and .
? Make p the subject of both and
? Eliminate p by equating both expressions to find the relationship between and .
SLIDE 28
When two objects collide, their combined mass is inversely proportional to their combined velocity . .
?
This quantity, p, is called momentum.
The velocity and mass of a moving object at one time are and . This object collides with another, and later, their combined mass and velocity are and .
? and
? Law of conservation of momentum in any collision
This may also be written
momentum before collision momentum after collision
SLIDE 29
Momentum
The product of the mass and velocity of an object is a constant called momentum
Law of conservation of momentum in any collision
momentum before collision momentum after collision
SLIDE 30
Three toy trucks, each with a mass of 2 kg are moving with velocity of towards another stationary truck, also with a mass of 2 kg. They collide and couple together. The four trucks continue to move together at a constant velocity.
What is the final velocity of the four trucks moving together?
SLIDE 31
Three toy trucks, each with a mass of 2 kg are moving with velocity of towards another stationary truck, also with a mass of 2 kg. They collide and couple together. The four trucks continue to move together at a constant velocity.
What is the final velocity of the four trucks moving together?
Solution
momentum before collision momentum after collision
SLIDE 32
Functions, map, mappings
A function maps one number to another.
This is read f of x is x squared.
Another way of writing this is
This is read, f maps x to x squared.
Both mean the same thing and are alternative ways of presenting the same idea.
Function, map and mapping are names for the same thing.
SLIDE 33
Complete the following table of values of the function for different arguments of x.
4
16 1
SLIDE 34
4
16 9 4 1 0 1 4 9 16
SLIDE 35
x
4
y 16 9 4 1 0 1 4 9 16
A function is represented by a graph. A graph is a picture of what a function does. For plot the points onto the graph below, and join the points with a smooth, continuous curve.
SLIDE 36
x
4
y 16 9 4 1 0 1 4 9 16
SLIDE 37
Parabola
The graph of is a parabola.
SLIDE 38
Graph of direct linear proportionality
The graph of a relationship of direct linear proportionality is a straight line through the origin. The gradient of the graph is the constant of proportionality.
Two variables, X and Y, are directly proportional. On the same paper, draw the graph of between the arguments to when ? , ? , ? .
Hint. The graphs are linear and proportional (straight lines through the origin). To draw such a line, you only need the origin and one other point. For each value of k, find Y when , and join this point to the origin.
SLIDE 39
? ? ?
SLIDE 40
Find the gradients of the following two lines, and hence express each relationship in the form , where k is a constant of proportionality.
SLIDE 41
?
?
SLIDE 42
Graph of inverse proportionality
Complete the following table of values for and plot its graph.
x 0.1 0.2 0.5 1 2 5 10
y 10 0.5
SLIDE 43
x 0.1 0.2 0.5 1 2 5 10
y 10 5 2 1 0.5 0.2 0.1
SLIDE 44
Rectangular hyperbola
The graph of is called a rectangular hyperbola, or just hyperbola for short.
SLIDE 45
We explore what happens to when x becomes larger and larger.
? Complete the following table
x 1 10 100 1000 10000 100000
y 1 0.1
? What happens to as x gets larger and larger?
? Will ever become 0?
SLIDE 46
? For
x 1 10 100 1000 10000 100000
y 1 0.1 0.01 0.001 0.0001 0.00001
? As x gets larger and larger, gets closer and closer to 0.
? Nonetheless, never becomes 0.
SLIDE 47
Limits
As x gets larger and larger, gets closer and closer to 0. Nonetheless, never becomes 0.
This is written
This is said, As x tends to infinity, tends to 0.
0 is called the limit of .
SLIDE 48
?
Write this out in words and explain what is happening.
? As x tends to infinity, tends to 0.
Write this using arrows.
SLIDE 49
?
As x tends to infinity, tends to 0.
Alternatively, As x gets bigger and bigger, gets closer and closer to 0.
? As x tends to infinity, y tends to 0.
SLIDE 50
x 1 10 100 1,000 10,000 100,000 1,000,000
y 1 0.1 0.01 0.001 0.0001 0.00001 0.000001
Sketch these points onto the following graph. The x-axis uses a logarithmic scale. You will not be able to draw the curve accurately as the points lie too close to the x-axis. Make a sketch.
SLIDE 51
Asymptote
x 1 10 100 1,000 10,000 100,000 1,000,000
y 1 0.1 0.01 0.001 0.0001 0.00001 0.000001
The curve gets closer and closer to the x-axis but never touches it.
Such a line to which a curve gets closer and closer is called an asymptote.
SLIDE 52
Asymptotes
The graph of is asymptotic to the x- and y-axes.
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