Question

Find constant of variation and write equation of variation for every example given below.
(1) p ∝ $\frac{1}{q}$ ; if p = 15 then q = 4
(2) z ∝ $\frac{1}{w}$ ; when z = 2.5 then w = 24
(3) s ∝ $\frac{1}{t^2}$ ; if s = 4 then t = 5
(4) x ∝ $\frac{1}{\sqrt{y}}$ ; if x = 15 then y = 9

Answer 1

(1)
$p\propto \frac{1}{q}$

$\therefore p=\frac{k}{q}$, where k is constant of variation

When p = 15, q = 4.

$$\begin{gathered} \therefore 15=\frac{k}{4} \\ \Rightarrow k=15\times 4=60 \end{gathered}$$
So, the equation of variation is $p=\frac{60}{q}$ or pq = 60.

(2)
$z\propto \frac{1}{w}$

$\therefore z=\frac{k}{w}$, where k is constant of variation

When z = 2.5, w = 24.

$$\begin{gathered} \therefore 2.5=\frac{k}{24} \\ \Rightarrow k=2.5\times 24=60 \end{gathered}$$
So, the equation of variation is $z=\frac{60}{w}$ or zw = 60.

(3)
$s\propto \frac{1}{t^2}$

$\therefore s=\frac{k}{t^2}$, where k is constant of variation

When s = 4, t = 5.

$$\begin{gathered} \therefore 4=\frac{k}{5^2} \\ \Rightarrow 4=\frac{k}{25} \\ \Rightarrow k=25\times 4=100 \end{gathered}$$
So, the equation of variation is $s=\frac{100}{t^2}$ or st² = 100.

(4)
$x\propto \frac{1}{\sqrt{y}}$

$\therefore x=\frac{k}{\sqrt{y}}$, where k is constant of variation

When x = 15, y = 9.

$$\begin{gathered} \therefore 15=\frac{k}{\sqrt{9}} \\ \Rightarrow 15=\frac{k}{3} \\ \Rightarrow k=15\times 3=45 \end{gathered}$$
So, the equation of variation is $x=\frac{45}{\sqrt{y}}$ or $x\sqrt{y}=45$.