2.9 Answers to the Additional Problems

1. $\partial z/\partial x=(0.00187y^{0.7})=$constant for given $y$.
   Exposed profile should be a straight line.
   $\partial z/\partial y=(0.00187)(x+1)(0.7)y^{-0.3}$
   Exposed profile curves upward with decreasing slope.

2. $\partial z/\partial x=2(P1)x$, $\partial z/\partial y=2(P2)y$
   $\partial^2 z/\partial x^2=2(P1)$, $\partial^2z/\partial x\partial y=0$, $\partial^2z/\partial y^2=2(P2)$
   $\partial z/\partial x=0$ if $x=0$
   $\partial z/\partial y=0$ if $y=0$
   Thus $(x,y)=(0,0)$ is the only critical point.
   To classify the critical point, evaluate: $$\frac{\partial^2z}{\partial x^2}\frac{\partial^2z}{\partial y^2}-\Big(\frac{\partial^2 z}{\partial x\partial y}\Big)^2=4(P1)(P2)$$ Pick $P1=.6$, $P2=1.0$
   Then $4(P1)(P2)>0$, $\frac{\partial^2z}{\partial x^2}=2(P1)>0$ and the point is a relative minimum.

3. We use the Pythagorean theorem to obtain

$$(2r)^2=W^2+D^2$$ Thus $$\begin{align*} D^2&=4r^2-W^2 \\ S&=(0.1)4r^2W-(0.1)W^3 \\ \frac{dS}{dW}&=(0.4)r^2-(0.3)W^2 \\ 0&=(0.4)r^2-(0.3)W^2 \end{align*}$$

The critical point is then at $W=\sqrt{4r^2/3}=2r/\sqrt{3}$ We have $$\frac{d^2S}{dW^2}=-0.6W<0$$ so that when $W=2r/\sqrt{3}$, $S$ is indeed at a maximum.
The depth is then $$\begin{align*} D&=\sqrt{4r^2-W^2} \\ &=\sqrt{4r^2-4r^2/3} \\ &=2\sqrt{2/3}r \end{align*}$$

4.a. Even assuming $r = r(t)$, N cannot exceed $K$ if $N(0) < K$. We see this by writing $N$ as a fraction. $$N(t)=\frac{K}{1+be^{-r(t)t}}$$ b. Treat $r(t)$ in the difference equation model, with $$\begin{align*} r(0)&=r_0 \\ r(1)&=r_0/2 \\ r(2)&=r_0/3 \\ r(3)&=r(2) \end{align*}$$ or some such scheme to decrease r as time increases.

5. Evaluate the derivative. $$Y'=(P1)X-(P2)X^2$$ The “rate of increase” of $Y$ is greatest when $Y'$ is maximal. Find, the max($Y'$) by differentiating $Y'$ and setting $Y'' = 0$. $$\begin{align*} Y''&=\frac{d(Y')}{dx}=(P1)-(2)(P2)X \\ 0&=(P1)-(2)(P2)X \\ X&=(P1)/(2(P2)) \end{align*}$$ The value for $Y$ is then $$Y=\Big[\frac{P1}{2(P2)}\Big]^2\Big[\frac{P1}{2}-\frac{(P1)(P2)}{6(P2)}\Big]=\frac{(P1)^3}{12(P2)^2}$$ This point is an inflection point on the graph $Y$ versus $X$.