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Models of physical situations are often arrived at by recording the values
of variables in a table and by sketching graphs. Examining the tabulated
values and the graph, we try to define the relationship between the
variables by an equation. The equation thus derived is called a
mathematical model.
If the value of a dependent quantity changes as a result of changes in
the value of an independent quantity, then we say that one quantity varies
with respect to the other. However, the type of variation
depends on the relationship.
Direct Proportion (or Variation)
A cyclist's progress over a journey of 120 km is recorded. The results
are tabulated below and plotted on a graph.
In such a case, we say that:
This is written as:
From the tabulated values, we find that:
In general:
The constant k is called the constant of variation or the
constant of proportionality.
Graphically, this relation takes the shape of a straight line with
gradient equal to the value of k.
Example 9
If y varies directly as x and y = 15 when x
= 5, find the formula
connecting x and y. Hence find y
when x = 8.
Solution:
This is the required formula.
Example 10
Use the following diagram to find the relationship that exists between x
and y. Hence find y when x is 9.5.
Solution:
Key Terms
mathematical model, variation,
direct proportion, direct variation,
constant of proportionality, constant
of variation
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