# Suppose r varies directly as the square of m, and inversely as s. If r equals 14 when m equals 8...

## Question:

Suppose {eq}r {/eq} varies directly as the square of {eq}m {/eq}, and inversely as {eq}s {/eq}. If {eq}r {/eq} equals {eq}14 {/eq} when {eq}m {/eq} equals {eq}8 {/eq} and {eq}s {/eq} equals {eq}4 {/eq}, find {eq}r {/eq} when {eq}m {/eq} equals {eq}24 {/eq} and {eq}s {/eq} equals {eq}4 {/eq}.

## Proportionality:

In this problem we are going to find r given m and s.

r is directly proportional to square of m => r increases as square of m increases.

r is inversely proportional to s => r increases as s decreases.

Given ,

{eq}\begin{align*} r & \propto m^2 \space , r \propto \frac{1}{s} \\ => \ r & \propto \frac{m^2}{s} \\ => \space r &= (K)\cdot \frac{m^2}{s} &\text{[ Where K is proportionality constant.]} \\ &\text{ Given r =14 when m = 8 and s = 4 } \\ => \space 14 &= (K) \cdot \frac{8^2}{4} \\ => \space K &= \frac{14}{16} = \frac{7}{8} \\ => \space r &= (\frac{7}{8}) \cdot \frac{m^2}{s} \\ => \space r \bigg |_{m = 24 , s = 4 } &= (\frac{7}{8} )\cdot \frac{24^2}{4} \\ &= (7) \cdot (18) = 126 \end{align*} {/eq}