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 |