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Section 2.6 #15: The time $t$ required to drive a fixed distance varies inversely as the speed $r$. It takes $5$ $\mathrm{hr}$ at a speed of $80$ $\dfrac{\mathrm{km}}{\mathrm{hr}}$ to drive the fixed distance. How long will it take to drive the same distance at a speed of $70$ $\dfrac{\mathrm{km}}{\mathrm{hr}}$?
Solution: From the first sentence we know the following equation holds for some $k$:
$$t = \dfrac{k}{r}.$$
Plugging in $t=5$ and $r=80$ we get the qeuation
$$5 = \dfrac{k}{80},$$
so we may solve for $k$ by multiplying by $80$ to get
$$400=k.$$
This means
$$t = \dfrac{400}{r}.$$
We are asked to find $t$ given that $r=70$. To do this plug in $r=70$ into the equation to see that
$$t = \dfrac{400}{70} \hspace{2pt} \mathrm{hr}.$$
Section 2.6 #17: The maximum number of grams of a fat that should be in a diet varies directly as a person's weight. A person weighing $120$ $\mathrm{lb}$ should have no more than $60$ $g$ of fat per day. What is the maximum daily fat intake for a person weighing $180$ $lb$?
Solution: Let $G$ denote the number of grams of fat and let $W$ denote weight. From the first sentence we get the equation, for some $k$,
$$G=kW.$$
Plugging in $G=60$ and $W=120$ yields
$$60=120k,$$
and solving for $k$ yields
$$k=\dfrac{60}{120} = \dfrac{1}{2}.$$
Therefore the equation is
$$G = \dfrac{W}{2}.$$
To answer the question, we plug in $W=180$ to get
$$G = \dfrac{180}{2} = 90,$$
so a 180 $lb$ person should eat at most $90$ grams of fat in a day.
