Chapter 1, Problem 134RE

To determine

To calculate:The equation that depicts $z$ varies inversely with respect to $y$ and also, $z=12$ and $y=16$

Answer to Problem 134RE

The required equation is $z=\frac{192}{y}$

Explanation of Solution

Given information:

Here, $z$ varies inversely with respect to $y$ and also, $z=12$ and $y=16$

Formula used:

For 2 variables say, $x$ and $y$ , the statement $x$ is directly proportional to $y$ can be written as:

  $x\alpha y$

Which can be written as:

  $x=ky$

Where $k$ denotes the proportionality constant.

Similarly the statement $x$ is inversely proportional to $y$ can be interpreted as:

  $x\alpha \frac{1}{y}$

Which can be written as:

  $x=k\left(\frac{1}{y}\right)$

Where $k$ denotes the proportionality constant.

Slope-intercept equation for a given line which has slope as $m$ and $y$ −intercept as $b$ is:

  $y=mx+b$

Calculation:

As $z$ varies directly with respect to $y$ .

Recall, For 2 variables say, $x$ and $y$ , the statement $x$ is inversely proportional to $y$ can be interpreted as:

  $x\alpha \frac{1}{y}$

Which can be written as:

  $x=k\left(\frac{1}{y}\right)$

Where $k$ denotes the proportionality constant.

Hence, this variation can be expressed as follows:

  $\begin{array}{l}z\alpha \frac{1}{y}\\ z=\frac{k}{y}\end{array}$ $\left(1\right)$

Where $k$ denotes the proportionality constant.

It is also given $z=12$ and $y=16$

Put these values in $\left(1\right)$ to get:

  $\begin{array}{l}12=\frac{k}{16}\\ k=12*16\\ k=192\end{array}$

Replace the value of $k$ in $\left(1\right)$ to get the required equation as:

  $z=\frac{192}{y}$

Thus, the required equation is $z=\frac{192}{y}$