1. Problem: Write equations for the given relationships. - w is directly proportional to v. - a varies inversely with b. - r is inversely proportional to s cubed. - m varies directly with n squared. Formulas: - Direct variation: $y = kx$ - Inverse variation: $y = \frac{k}{x}$ 1. Since w is directly proportional to v, the equation is: $$w = kv$$ where $k$ is the constant of proportionality. 2. a varies inversely with b, so: $$a = \frac{k}{b}$$ 3. r is inversely proportional to $s^3$, so: $$r = \frac{k}{s^3}$$ 4. m varies directly with $n^2$, so: $$m = kn^2$$ --- 2. Determine if the equation represents direct or inverse variation and find the constant. 5. $t = \frac{100}{s}$ - This is inverse variation because $t$ varies as $\frac{1}{s}$. - Constant $k = 100$. 6. $\frac{y}{5} = x$ or $y = 5x$ - This is direct variation. - Constant $k = 5$. 7. $C = 2\pi r$ - This is direct variation between $C$ and $r$. - Constant $k = 2\pi$. 8. $xy = \frac{2}{5}$ - This is inverse variation between $x$ and $y$. - Constant $k = \frac{2}{5}$. --- 3. Use variation models to find missing values. 9. $y$ is directly proportional to $x$, $y = kx$. Given $y=17.5$ when $x=21$: $$k = \frac{y}{x} = \frac{17.5}{21} = \frac{5}{6}$$ Find $y$ when $x=39$: $$y = kx = \frac{5}{6} \times 39 = 32.5$$ 10. $m$ varies directly with $n$, so $m = kn$. Given $m=209$ when $n=22$: $$k = \frac{209}{22}$$ Find $n$ when $m=361$: $$361 = k n \Rightarrow n = \frac{361}{k} = \frac{361}{\frac{209}{22}} = 361 \times \frac{22}{209} = \frac{7942}{209} = 38$$ 11. Cost varies directly with number of copies. Given cost $C=45$ for 750 copies. Find cost for 750 + 120 = 870 copies. $$k = \frac{45}{750} = 0.06$$ Cost for 870 copies: $$C = 0.06 \times 870 = 52.2$$ 12. Time varies inversely with rate. Given time $t=45$ min at rate $r=8$ gal/min. Find time $t_2$ at rate $r_2=15$ gal/min. $$t = \frac{k}{r} \Rightarrow k = tr = 45 \times 8 = 360$$ $$t_2 = \frac{k}{r_2} = \frac{360}{15} = 24$$ Final answers: 9. $y=32.5$ 10. $n=38$ 11. Cost = 52.2 12. Time = 24 minutes