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

Assignment 12

Problems

Racing in the Bay

A sailboat race is being planed in the Bay, and you are hired as a data scientist to consult with the racing coaches to plan routes and ensure fairness. The planned race starts in the Mission Bay Neighborhood near UCSF and ends in Alameda on the Oakland shore; but there is a problem: this crosses directly under the Bay Bridge, and so the sailboats will have to route around the pillars supporting the suspension bridge out in the bay. Because this crosses a busy shipping lane, the Bay Transit Authority requires racers to be flexible, and take their sailboats on either the eastern or western side of the obstructing pillar depending on boat traffic. To comply with this, the race team has set out two paths across the bay, one on each side of the pillar, as depicted in the map below.

Not all sailboats will necessarily take the same path (depending on shipping traffic different boats may be routed along different paths, and even in the absence of external traffic the race organizers may prefer to spread boats across the paths to decrease congestion), so it is naturally important to quantify their differences. Simplified mathematical models for the two basic trajectories are given below (in units convenient for the problem, so \(1\) represents the width of the bay)

\[\mathrm{North Route}: n(t)=(t^2,t^3)\,t\in[0,1]\] \[\mathrm{South Route}: s(t)=(t,t^2)\, t\in[0,1]\]

The obvious first concern the race planners ask you about is whether the distances are fair. Sailboats don’t follow a route exactly and your research shows you that any given boat differs from the idealized route by an average of 7 percent of its length, so as long as the routes are within this margin of error they count as fair. You remember that its easy to find the length of a path using an arclength integral, and set off to integrate \(ds\) along these two paths. I’ve done the southern route for you \[\mathrm{South Length}= \int_0^1 \|s^\prime\|dt = \int_0^1 \sqrt{1+4t^2}dt = 1.4798\]

QUESTION I: Compute the length of the northern route. Should you approve these two routes as valid for the race? (That is, do they differ less than the acceptable error?)

The next concern of the race planners has to do with the tides. The San Francisco Bay is famous for rapid tides, as the entire volume of the bay is connected to the sea by only the small strait of the Golden Gate (this was one motivation for building Alcatraz out here - the quick tides make escape by swimming difficult). These fast tides can have a rather significant effect on the race as boats traverse the waters, and so a second question of fairness arises: does the flow of water give either the north or south route a distinct advantage in the race? Conditions vary on a day-by-day basis so the race commission cannot hope for an exact answer, but the main thing they have hired you to do is provide a reliable recommendation on fairness here. You download data for tidal patterns matching the race date/time over the past several years and produce from them an average to serve as your theoretical baseline water flow, given by the vector field \[\vec{W}=\left\langle 2xy-y, x^2-x+1\right\rangle\]

The net affect of the tides on a boat traveling along a curve \(C\) is the line integral \(\int_C\vec{W}\cdot d\vec{s}\). The race is considered fair if the net affect between different routes differs by less than 7 percent (the previously agreed-to margin of error).

QUESTION II: Are these two paths fair, after considering the effect of the tide flow? Do you recommend the race proceed, or a new plan be drawn up?

Power of a Windmill

For this problem you are working for a green energy company aiming to install a new wind farm in the Bay Area. A windmill generates its power by ‘stealing’ a fraction of the winds energy: this fraction is called the efficiency of the windmill design, denoted \(\varepsilon\), and typically ranges between \(20\) and \(40\) percent. The power available in the wind itself is calculated from the flow of air through the surface swept out by the windmill. That is, if the blades sweep out a surface \(S\) in space and the wind is given by the velocity vector field \(\vec{W}\), then the power is given by a surface integral

\[\mathrm{Power}= \varepsilon\,\left|\iint_S \vec{W}\cdot \hat{N}\,d A\right|\]

Where \(\hat{N}\) is the unit normal vector in the direction the windmill is facing, and \(dA\) is the infinitesimal area element for the surface \(S\).

The area swept out (red), the normal vector (yellow) and the wind vector field (blue).

Your company is considering two different possible scenarios, and has tasked you with making a recommendation to upper management. The two scenarios are described below:

  • (The Coastal Plan) Windmills will be built out at sea, approximately eight miles off the western coast of San Francisco (they were moved this far out after complaints were lobbied at a previous iteration of the plan that had them near shore, which would obstruct the view. After that meeting the plan was revised by calculating how far offshore they need to be so they are below the horizon for beachgoers, and this was the result). The windmills will face directly west to take advantage of the ocean breeze. They will be large and tall, but there will be relatively few of them, as they are out to sea and difficult to maintain. Below are some more precise stats about the plan

    • A coastal windmill has a blade length of 150 meters. We can model the area swept out by such a windmill facing west as a disk in the \(xz\) plane of this radius, centered at \((0,0,0)\) with normal vector pointed along the \(y\) axis.
    • In these coordinates, the expected windflow out at sea is modeled by the vector field \(\vec{W}(x,y,z)=\langle xz+y, 3+x,xyz-y^2\rangle\).
    • The efficiency of this large design is \(36\) percent.
    • The wind farm will contain 50 identical windmills.

  • (The East Bay Plan) Windmills will be built in the hills of the east bay, in open corridors between the mountains to maximize their ability to capture wind which has been trapped and funneled into straight gusts. These windmills will face directly north to capture these channeled breezes making their way inland from the north bay. These land-based windmills will be smaller, but the company plans to utilize more of them as they are cheaper to construct and maintain, so can have more for the same budget. Below are more precise stats on this plan:

    • A land based windmill has a blade length of 100 meters. We can model the area swept out by such a windmill facing north as a disk in the \(yz\) plane of this radius, centered at \((0,0,0)\) with normal vector pointed along the positive \(x\) axis.
    • In these coordinates, the expected windflow in the mountain valley is modeled by the vector field \(\vec{W}(x,y,z)=\langle 4+x+y,xyz,yz-2x\rangle\).
    • The efficiency of this smaller design is \(28\) percent
    • The wind farm will contain 125 identical windmills.

QUESTION: Your boss wants to build whichever of the plans will produce the most electrical power. Do some calculations, and then give them a recommendation.

Hints to setting it up! You want to do a surface integral, so you need to (1) figure out what the wind vector field is on the surface you care about. That means plugging in any value you know for \(x,y,\) or \(z\) defining the surface. (2) You need to then take the dot product of this with the normal vector to the surface, to get the flux you will integrate. (3) You need to write down a double integral of this: and perhaps change coordinates to make it easier?

Course Evaluations!

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