Problem $7.$ The strength $S$ of a rectangular wooden beam is proportional to its width times the square of its depth.

$($a$)$ Find the dimensions of the strongest beam that can be cut from a $12-$centimeters diameter cylindrical log$.$ Demonstrate your results using diagrams.

$($b$)$ Assuming the proportionality constant to be $k=1,$ graph $S$ as a function of the beam's depth $d,$ again taking $k=1.$ Compare the graph with your answer in part $($a$).$

Problem $8.$ A customer has asked you to design an open$-$top rectangular stainless steel vat. It is to have a square base and a volume of $32f{t}^3,$ welded from a quarter$-$inch plate, and weigh no more than necessary. What dimensions do you recommend?

Problem $9.$ When we cough, the trachea $($windpipe$)$ contracts to increase the velocity of the air going out. This raises the question of how much it should contract to maximize the velocity and whether it contracts that much when we cough. Under reasonable assumptions about the elasticity of the tracheal wall and about how the air near the wall is slowed by friction, the average velocity $v$ can be modeled by the equation

$v(r)=c(r_0-r)r^{2}c\frac{m}{sec,\frac{r_0}{2}}r{r}_0$

where $r_0$ is the resting radius of the trachea in centimeters and $c$ is a positive constant whose value depends in part on the length of the trachea.

$($a$)$ Show that $v$ is maximum when $r=(\frac{2}{3})r_0$; that is$,$ when the trachea is about $33\%$ contracted.

$($b$)$ Considering the values $r_0=0\mathrm{.}5$ and $c=1,$ graph $v$ over the interval $0r0\mathrm{.}5.$ Compare what you see with your results in part $($a$).$