$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\length}{\operatorname{length}} \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\erf}{\operatorname{erf}} \newcommand{\ihat}{\hat{\imath}} \newcommand{\jhat}{\hat{\jmath}} \newcommand{\khat}{\hat{k}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)} $$

Assignment 5

Problems

Reading Contour Plots

You are hired as a consultant for an aerospace company that is working to design the next generation of rocket boosters. The main goal of your team is to improve the of the engine (the total impulse delivered per kilogram of fuel). You begin by focusing on the two most important variables: the rocket engine nozzle size \(n_s\), and the temperature \(T\) of the burning fuel. You write a computer program to simulate thousands of different combinations of temperature and size, and compute the efficiency \(E_{ff}(T,n_s)\) of each configuration. After a lot of computation, your program outputs the following figure:

A multivariate function of efficiency for rocket engine design.

How do you summarize your findings to your boss, given this data? An example summary might sound something like this (though this is not the answer for this exact plot, of course!)

For the most part, rocket engines perform better when the nozzle size is large and the temperature is small, though if the temperature gets too low it starts performing worse again. Performance is particularly bad for temperature of 300 and small nozzle sizes (around 1). The most efficient configuration is nozzle=5 and temp=3.2k.

If the current rocket engines in production have a nozzle size of 1 and a fuel temperature of 6k, what concrete suggestion could you make to your team on how to change the design slightly to improve efficiency? What would you say to a team currently building a rocket with nozzle size 7 and temperature 3k?

Example questions you may want to consider: Should you increase both temp and nozzle size? Decrease one of them and leave the other constant?Decrease both? Should they make big or small changes to their current design? Etc….

Partial Derivatives

The data of multivariate functions can be presented to us in multiple ways. Sometimes, we are given a contour plot - for example, as the output of a simulation like in the problem above. But other times we are just given an equation, and have to work with that.

In this problem we will continue our roll as an aerospace consultant engineer, working on the same rocket design as above. After successfully helping with the engine design, you are asked to work closer with the team that is building the rocket’s body so that you can safely integrate the engine. This team is run by a collection of theorists who have been able to work out a model for the rocket’s aerodynamic efficiency in terms of the angle \(\alpha\) and height \(h\) of its fins, the radius \(r\) of the main tube, and the thickness \(t\) of the metal support beams. Their model is

\[E = re^{-r/2}\sin(h \alpha / t)\]

Because of your success in optimizing the engine design, this team asks you for your help in optimizing the rocket body. In particular, they are currently considering building a rocket body with radius \(1.5\), beam thickness of \(0.3\), fin angle of \(\pi/4\) and fin height of \(1\).

They want to know should they increase or decrease the beam thickness, to improve efficiency? Because the rocket body design depends on so many parameters, we cant draw its graph (which lives in 5 dimensional space) or its contour plot (which lives in four dimensions!). So, we need to just do some calculus.

Graphs of Multivariate Functions

You are working with a real-estate developer to plan new environmentally-friendly infrastructure for a community in the rolling hills of Marin county. One particular hill you are developing is well approximated by the graph of the quadratic function \[z=5-\frac{x^2}{4}-\frac{y^2}{8}\] For \((x,y)\in [-2,2]\times[-2,2]\).

Your job is to find the optimal placement along the hill for the collection of solar panels that will provide electricity for the development. Solar panels work best when the sun light shines directly perpendicularly onto their surface, so you are looking for the location on the hill where the sun hits it most perpendicularly.

Consulting a local meteorologist you learn the direction of the average noontime sun at your build site is \(\langle 1,0,-4\rangle\). At which point \((x,y,z)\) should we build the solar panel farm? To figure this out, follow the outline below.

Finding the optimal location on a hill to place solar panels requires understanding the sun’s direction.

• At any point \((x,y)\) on the plane, can you use partial derivatives to find a vector which is perpendicular to the hill at that point?
• Our real goal is to find when the hill is perpendicular to the direction of the sun. For which \((x,y)\) is the vector you found parallel to this?

Solutions

Reading Contour Plots

This plot shows that rocket engine efficiency is maximized around a nozzle size of 2, and a temperature of 5k. Small changes from this configuration in any way decrease the efficiency somewhat. The worst performing rocket engines are those with large nozzle size, and temperature around mid-range (3k degrees).

If the team is currently building a rocket with nozzle size 1 and temperature 6k, they should slightly decrease the temperature while increasing nozzle size, in order to best increase efficiency.

Partial Derivatives

A function is increasing if its derivative is positive, and decreasing when the derivative is negative. Here we want to know if the efficiency is increasing or decreasing as we change the beam thickness, so we need to take the partial derivative in the direction of the variable \(t\).

Computing this,

\[\partial_t E=\partial_t\left(re^{r/2}\sin\left(\frac{h\alpha}{t}\right)\right)=re^{-r/2}\cos\left(\frac{h\alpha}{t}\right)\left(\frac{-h\alpha}{t^2}\right)\]

Next, because we are only interested in what is happening near the actual rocket configuration being built, we must plug in the current rocket parameters of \(r=1.5, t=0.3\) and \(\alpha=\pi/4\) to get

\[\begin{align*}\partial_t E&=1.5 e^{-0.75}\cos\left(\frac{(1)(\pi/4)}{0.3}\right)\left(\frac{-(1)(\pi/4)}{0.3^2}\right)\\ &=0.70854\cdot (-0.8660)\cdot (-8.7266)\\ &=5.354603 \end{align*}\]

This is positive, so we should increase thickness if we want to improve the efficiency of our current rocket design.

In fact this tells us even more quantitative information. Since this value \(5.35\ldots\) is the slope this says if we increase thickness by a small amount like \(0.1\) then the efficiency increases by \(5.35\cdot 0.1 = 0.535\).

Graphs of Multivariate Functions

In this problem we know the direction of the sun’s rays, and we want to know when the normal vector to the hill is parallel to this: that is when is the hills normal vector a scalar multiple of \(\langle 1,0,-4\rangle\)?

The first step is to find the normal vector to the hill \(z=5-\frac{x^2}{4}-\frac{y^2}{8}\) at the point \((x,y)\). In class, we found the normal vector to the graph of a multivariate function to be the coefficient vector from its tangent plane:

\[n=\langle f_x,f_y,-1 \rangle\]

This version is downward pointing as the \(z\) component is \(-1\), so looking at our picture, this normal vector is pointed into the hill.

Alternatively - if we did not remember this we could derive it using our multivariate tools of partial differentiation and the cross product! Here we could find two vectors that are tangent to the hill using partial derivatives: if you move infinitesimally one unit in the \(x\) direction your \(z=f(x,y)\) component will move by \(\partial_x f\), and so the vector \[v_x = \pmat{1\\ 0 \\ f_x}\] is tangent in the \(x\)-direction. Similarly, \(v_y=\langle 0,1 f_y\rangle\) is tangent in the \(y\) direction, and their cross product is normal: \[n=v_x\times v_y=\left| \begin{matrix}\ihat &\jhat&\khat\\ 1&0& f_x\\ 0&1& f_y \end{matrix} \right|=\pmat{-f_x \\ -f_y\\ 1}\] This normal is upward pointing at \(z\) is positive: looking at our picture its pointed out of the hill.

To find the actual vector for this problem, we need to calculate these derivatives:

\[f_x = -\frac{x}{2}\hspace{1cm}f_y=-\frac{y}{4}\] \[n=\left\langle\frac{-x}{2},\frac{-y}{4},-1\right\rangle\]

To figure out when this is a multiple of \(\langle 1,0,-4\rangle\) we are just asking when there is a single constant \(c\) such that

\[k\pmat{-x/2\\ -y/4 \\ -1}=\pmat{1\\ 0 \\ -4}\]

This is a system of three equations that we need to solve. The bottom equation tells us that \(k = 4\). Substituting this into the second equation, we see \(-ky/4 = 0\) implies \(y=0\), and into the first equation yields \[-kx/2 = -4x/2=2x=1\hspace{0.25cm}\implies\hspace{0.5cm}x=-1/2\]

Thus the point where the normal vector is parallel to the sunrays - where we should place our solar panel on the hill has \((x,y)\) coordinates \((-1/2,0)\).