Direct Variation With Exponents
Direct Variation With Exponents
Direct Variation:
Two terms are said to be in direct variation if increase or decrease of term will result in the increase or decrease of the other term respectively.
Direct Variation With Exponents:
Two terms are said to be in direct variation with exponents if increase or decrease of term will result in the exponential increase or decrease of the other term respectively.
For example:let us consider the formula to find the area of circle.
The formula for finding the area A of a circle of radius r is
Now let us calculate the value of area for different values of the radius
|
r |
0 | 1 | 2 | 3 | 4 |
|
A |
0 | 3.14 | 12.57 | 28.27 | 50.27 |
If we graph A against r we get the graph below
The graph is not the straight line, it is curve or we can say that it is part of a parabola . So the area is not directly proportional with radius. A is not directly proportional to r.
However suppose we include in the table a row for the values of :
|
r |
0 | 1 | 2 | 3 | 4 |
| 0 | 1 | 4 | 9 | 16 | |
| A | 0 | 3.14 | 12.57 | 28.27 | 50.27 |
The graph of A against is a straight line through the origin O as shown below
so A is directly proportional to . We can say that
. In this case we know from the formula that the proportional constant is
.
Notice from the table that if r is doubled from 1 to 2 both and A are multiplied by 9 .
Illustration: From the following equation state which two variables are directly proportional and determine the proportionality constant k .
.
Solution: On analysing the equation we can say that as
,
.
Hence y is directly proportional to the fourth power of x and the proportionality constant is
.
If p varies directly as the square of q, and p = 20 when q = 5, find p when q = 8. | |||
| Right Option : A | |||
| View Explanation | |||
The distance a body falls from rest varies directly as the square of the time it falls (ignoring air resistance). If a ball falls 144 feet in three seconds, how far will the ball fall in seven seconds? | |||
| Right Option : A | |||
| View Explanation | |||
From the following equation state which two variables are directly proportional. | |||
| Right Option : D | |||
| View Explanation | |||
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