Question

Suppose that $y$ varies jointly with $w$ and $x$ and inversely with $z$ and $y=360$ when $w=8$, $x=25$ and $z=5$. How do you write the equation that models the relationship. Then find $y$ when $w=4$, $x=4$ and $z=3$?

Answer 1

Finding the equation that models the relationship between $y$, $w$, $x$ and $z$:

Step 1: Construction of the equation:

If a quantity $p$ varies jointly with the quantities $a$ and $b$ and inversely with $c$, then we get the following:

$$\begin{gathered} p\propto \frac{a.b}{c} \\ \Rightarrow p=\frac{k.a.b}{c} \end{gathered}$$

where, $k$ is the constant of variation.

Here, in this question, $y$ varies jointly with $w$ and $x$ and inversely with $z$, so, using the above formula,

we get, $y=\frac{k.w.x}{z}$, where $k$ is the constant of variation.

Step 2: Find the value of $k$:

Given that, $y=360$ when $w=8$, $x=25$ and $z=5$. So, putting $y=360$, $w=8$, $x=25$ and $z=5$ in $y=\frac{k.w.x}{z}$, we obtain:

$$\begin{gathered} y=\frac{k.w.x}{z} \\ \Rightarrow 360=\frac{k.8.25}{5} \\ \Rightarrow k=\frac{360\times 5}{200} \\ \therefore k=9 \end{gathered}$$

Step 3: Finding the value of $y$ when $w=4$, $x=4$ and $z=3$:

Substituting, $w=4$, $x=4$, $z=3$ and $k=9$ in $y=\frac{k.w.x}{z}$, we get the required value of $y$ as:

$\begin{array}{rcl}y& =& \frac{9\times 4\times 4}{3}\\ \therefore y& =& 48\end{array}$

Hence, the value of $y=48$.