Question

The time it takes you to go a certain distance varies inversely with the speed of travel. Suppose it takes you 4 hours to get to a sports event driving at a rate of 50 miles per hour.
a) Find the constant of variation. Write the particular equation expressing time in terms of speed.




b) How long will it take you to get to the event if you drive at 40 mph?

Answer 1

$\bf \qquad \qquad \textit{inverse proportional variation} \\\\ \textit{\underline{y} varies inversely with \underline{x}}\qquad \qquad y=\cfrac{k}{x}\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array}\\\\ -------------------------------$

$\bf \textit{time it takes varies inversely with the speed rate}\qquad \stackrel{time}{t}=\cfrac{k}{\stackrel{rate}{r}} \\\\\\ \textit{we also know that } \begin{cases} t=\stackrel{hrs}{4}\\ r=\stackrel{mph}{50} \end{cases}\implies 4=\cfrac{k}{50}\implies 200=k \\\\\\ \boxed{t=\cfrac{200}{r}} \\\\\\ \textit{if your rate r = 40, what is your distance \underline{t}?}\qquad t=\cfrac{200}{40}$