91Ó°ÊÓ

Blood Alcohol Level The percentage of alcohol in a person's bloodstream \(t\) hr after drinking 8 fluid oz of whiskey is given by $$ A(t)=0.23 t e^{-0.4 t} \quad 0 \leq t \leq 12 $$ How fast is the percentage of alcohol in a person's bloodstream changing after \(\frac{1}{2} \mathrm{hr}\) ? After \(8 \mathrm{hr}\) ? Source: Encyclopedia Britannica.

Two curves are said to be orthogonal if their tangent lines are perpendicular at each point of intersection of the curves. In Exercises \(89-92\), show that the curves with the given equations are orthogonal.$$ y-x=\frac{\pi}{2}, \quad x=\cos y $$

Two families of curves are orthogonal trajectories of each other if every curve of one family is orthogonal to every curve in the other family. In Exercises \(93-96\), (a) show that the given families of curves are orthogonal to each other, and (b) sketch a few members of each family on the same set of axes. $$ x^{2}+y^{2}=c^{2}, \quad y=k x, \quad c, k \text { constants } $$

Two families of curves are orthogonal trajectories of each other if every curve of one family is orthogonal to every curve in the other family. In Exercises \(93-96\), (a) show that the given families of curves are orthogonal to each other, and (b) sketch a few members of each family on the same set of axes. $$ 2 x^{2}+y^{2}=c, \quad y^{2}=k x, \quad c, k \text { constants } $$

A 20 -ft ladder leaning against a wall begins to slide. How fast is the angle between the ladder and the wall changing at the instant of time when the bottom of the ladder is \(12 \mathrm{ft}\) from the wall and sliding away from the wall at the rate of \(5 \mathrm{ft} / \mathrm{sec} ?\)