Chapter 8, Problem 5CT

Solution

To find: Whether the relationship between the variables is direct variation or an inverse variation, or neither. If the relation exists, write the equation to model the variation.

The relation between the variables is indirect variation. And the equation for indirect variation is written as $xy=128$ .

Given information:

The table is:

$x$	$y$
4	32
8	16
16	8
32	4

Concept used:

  $\left(a\right)$ Inverse variation: When one of the given variables increases with decrease in other variable, then it might be an inverse variation or vice versa provided the product of variables should remain constant. Mathematically, the product of $xy=k$ where $\left(k\right)$ is a constant of variation.

  $\left(b\right)$ Direct variation: When one variable increases with increase in other variable, or decreases with decrease in other variable then it might be a direct variation. The equation to model the direct variation is $\frac{y}{x}=k$ where $\left(k\right)$ is a constant of variation.

Calculation:

From the values of table, it can be interpreted that as increase in $\left(x\right)$ results decrease in $\left(y\right)$ . Therefore, using clause $\left(a\right)$ of concept stated above for indirect variation, the table of values might be in an indirect variation.

Therefore evaluate constant of variation $\left(k\right)=xy$ for each row from table and check whether the constant of variation $\left(k\right)$ is same for all or not.

$x$	$y$	$k=xy$
4	32	128
8	16	128
16	8	128
32	4	128

From the above table, it can be seen that the constant of variation $\left(k\right)$ is same and as $\left(x\right)$ increases, there is a decrease in $\left(y\right)$ . Hence, the relation between the variables is indirect variation. And the equation for indirect variation is written as $xy=128$ .

Conclusion:

The relation between the variables is indirect variation. And the equation for indirect variation is written as $xy=128$ .