Question

Suppose that $\mathrm{y}$ varies jointly with $\mathrm{w}$ and $\mathrm{x}$ and inversely with $\mathrm{z}$ and $\mathrm{y}=100$ when $\mathrm{w}=10$, $\mathrm{x}=25$ and $\mathrm{z}=5$. How do you write the equation that models the relationship?

Answer 1

Step-1: Find the equation between $\mathrm{x},\mathrm{y}\mathrm{and}\mathrm{z}$:

It is given that $\mathrm{y}$ varies jointly with $\mathrm{w}$ and $\mathrm{x}$:

.$\mathrm{y}\propto \left(\mathrm{w}\times \mathrm{x}\right)...\left(1\right)$

$\mathrm{y}$ varies inversely with $\mathrm{z}$ and this means:

$\mathrm{y}\propto \frac{1}{\mathrm{z}}...\left(2\right)$

Combining $\left(1\right)$ and $\left(2\right)$, we have:

$$\begin{gathered} \mathrm{y}\propto \frac{\left(\mathrm{w}\times \mathrm{x}\right)}{\mathrm{z}} \\ \mathrm{y}=\mathrm{c}\frac{\left(\mathrm{w}\times \mathrm{x}\right)}{\mathrm{z}}...\left(3\right) \end{gathered}$$

$\mathrm{c}$ denotes the coefficient of equality.

Step-2: Write the equation that models the relationship:

Substitute the known values in equation $\left(3\right)$:

$$\begin{gathered} 400=\mathrm{c}\times \frac{\left(10\times 25\right)}{5} \\ \Rightarrow 400=\mathrm{c}\times 50 \\ \Rightarrow \mathrm{c}=\frac{400}{50} \\ \therefore \mathrm{c}=8 \end{gathered}$$

Substitute the value of $\mathrm{c}$ in equation $\left(3\right)$:

$\mathrm{y}=8\frac{\left(\mathrm{w}\times \mathrm{x}\right)}{\mathrm{z}}$

Therefore, $\mathrm{y}=8\frac{\left(\mathrm{w}\times \mathrm{x}\right)}{\mathrm{z}}$is the equation that models the relationship.