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.

1. 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 $$X$$.