Question

if y variesbdirectly as x, and y is 400 when x is r and y is r when x is 4, what is the numeric constant of variation in this relation

Answer 1

$\bf \qquad \qquad \textit{direct proportional variation} \\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad y=kx\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}$

$\bf \textit{we know that } \begin{cases} y = 400\\ x = r&\qquad \qquad 400=kr\\[-0.5em] \hrulefill\\ y = r\\ x = 4&\qquad \qquad \boxed{r}=k4 \end{cases} \\\\\\ \stackrel{\textit{using substitution on the 1st equation}}{400=k\boxed{k4}\implies 400=4k^2}\implies \cfrac{400}{4}=k^2\implies 100=k^2 \\\\\\ \sqrt{100}=k\implies \blacktriangleright 10 = k \blacktriangleleft$