# If r varies inversely as t, but directly as the square of m. if r=32 when m=8 and t=2, find r when m=6 and t=5?

Feb 28, 2018

$r = 7.2$

#### Explanation:

r varies inversely as t, $\implies r \propto \frac{1}{t}$ ------(1)

but directly as the square of m $\implies r \propto {m}^{2}$ -------(2)

Combining (1) and (2):

$r \propto {m}^{2} / t$

writing it as a equation (removing proportionality sign):

$\implies r = k \times {m}^{2} / t$ ----(3),where $k$ is the proportionality constant.

$\implies k = r \times \frac{t}{m} ^ 2$

Given that if r=32 when m=8 and t=2, gives the value of $k$ as:

$\implies k = 32 \times \frac{2}{8} ^ 2 = \frac{64}{64} = 1$

$\therefore k = 1$------(1)

To find r when m=6 and t=5, substitute value of $k = 1$:

(3) $\implies r = 1 \times {6}^{2} / 5$

$\implies r = \frac{36}{5} = 7.2$

Feb 28, 2018

$r = \frac{36}{5}$

#### Explanation:

$\text{the initial statement is } r \propto {m}^{2} / t$

$\text{to convert to an equation multiply by k the constant}$
$\text{of variation}$

$\Rightarrow r = k \times {m}^{2} / t = \frac{k {m}^{2}}{t} \leftarrow \textcolor{b l u e}{\text{k is the constant of variation}}$

$\text{to find k use the given condition}$

$r = 32 \text{ when "m=8" and } t = 2$

$r = \frac{k {m}^{2}}{t} \Rightarrow k = \frac{r t}{m} ^ 2 = \frac{32 \times 2}{64} = 1$

$\text{equation is } \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{r = {m}^{2} / t} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{when "m=6" and "t=5" then}$

$r = \frac{36}{5}$