#### Different kinds of graphs.

1.1
a and b are directly proportional to each other because their graph is a straight line and the
Δb
gradient of the line is constant, so that the rate   ——   remains constant.
Δa
1.2
a and b are directly proportional to each other because their graph is a straight line (See 1.1).
1.3
a and b are directly proportional to each other because their graph is a straight line (See 1.1).
1.4
a and b are inversely proportional to each other because the product, ab, is a constant.
1.5
a and b are inversely proportional to each other because their product, ab, is a constant.
1.6
The relationship between a and b is unknown – probably it is a quadratic or cube relationship.
1.7
It is a quadratic relationship – a is equal to cb²   where c is a positive constant.
1.8
It is a quadratic relationship – a is equal to cb²   where c is a negative constant.
1.9
The relationship between a and b is unknown – probably it is a quadratic or cube relationship.
1.10
a and b are directly proportional to each other because their graph is a straight line (See 1.1).
1.11
a and b are directly proportional to each other because their graph is a straight line (See 1.1).
m has a negative value.
1.12
a and b are directly proportional to each other because their graph is a straight line (See 1.1).
cost
2.1
Yes. The rate   —————————   remains constant and it has a value of 4.
number of articles
2.2
The graph will be a straight line because the rate is costant – the variables are directly proportional to each other.
2.3
See the accompanying graph.
2.4
Yes, the shape is as predicted.
2.5.1
cost of 4 articles = 16 (P)
2.5.2
number of articles = 6,5 (Q)
2.6
No. A fraction of an article is not made.
men
3.1
No. The rate   ————   does not remain constant. The relationship is that men and hours
hours
are inversely proportional to each other because their product , men X hours, is a constant. 3.2
The graph will be a hyperbola because men and hours
are inversely proportional and their
product is a constant.
3.3
See the accompanying graph.
3.4
Yes.
3.5.1
9,6 hours. Yes, it is a suitable answer because one can
work for a fraction of an hour.
3.5.2
2,4 men. No, it is not a suitable answer because a fraction of a man does not exist.
3.6
48 hours, the time that 1 man will take to complete the task. Seeing that a fraction of a man
does not exist, it is the longest time.
3.7
Not propable. There will be 192 men necessary to complete the task and they will probably
be in one another's way. It is not a suitable solution.
b
4.1
Yes. The rate   ——   remains constant. The value is approximately 3 for every set of values.
a 4.2
The value of b is equal to three times the
value of a.
In symbols:   b   = 3a
4.3
The graph will be a straight line because a and b
are directly proportional to each other.
4.4
See the accompanying graph.
4.5.1
b = 18
4.5.2
a = 3,5
4.6
b = 18 and a = 3,5. The values correspond well.
b
5.1
No. The rate   ——   does not remain constant. The product, ab, is constant and therefore
a a and b are inversely proportional to each other.
5.2
a and b are inversely proportional to each other.
In symbols:   ab   =   16
5.3
The graph is a hyperbola.
5.4
See the accompanying graph.
5.5.1
b = 3,2 (P)
5.5.2
a = 1,6 (Q)
5.6
a = 1,6 and b = 3,2
b
6.1
No. The rate   ——   is not constant. Furthermore, the product, ab, is not constant
a
and therefore a and b are not inversely proportional to each other. b is equal to a²
The relationship between a and b is thus a quadratic relationship. 6.2
b is equal to the square of a [or a squared].
In symbols:   b = a².
6.3
The graph will be a parabola in the first quadrant.
6.4
See the accompanying graph.
6.5.1
b = 2,25 (P)
6.5.2
a = 3 (Q)
6.6
a = 3 and b = 2,25. Values correspond well.
7.1
Length and breadth are inversely proportional to each other because the product,
In symbols   :   lb = 36 . . . where l = length and b = breadth
7.2
The graph will be a rectangular hyperbola in the first quadrant because length and breadth
are inversely proportional to each other and both are positive.
7.3
 Length 1 2 4 6 9 12 18 36 Breadth 36 18 9 6 4 3 2 1 7.2
See the accompanying graph.
7.3.1
b = 12 (P)
7.3.2
l = 1,8
7.4
R is the point (6 ; 6)
7.5
At point R the x- and the y-coordinates
are equal, so that length = breadth.
7.6
Length of the sides of the square is 6 m.
Length = breadth, see 7.5 above.
7.7
Length of wire netting = perimeter of the square
= 4 x 6 m = 24 m
7.8
Perimeter of coop is 24,4 m and the area = 36 m².
Thus, 2(l + b) = 24,4 . . . (1)    and lb = 36 . . . (2)
From (1) : l =   12,2 — b
Put into (2): (12,2 — b)b = 36 so that b² —12,2b + 36 = 0
Thus b = 5 or b = 7,2 . . . solved by using the formula
Thus, breadth = 5 m and length = 7,2 m   OR   breadth = 7,2 m and length = 5 m