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Assignment 9

Problems

Optimizing Volume of a Wineglass

We so far have been using triple integrals to compute the total amount of some function \(f\) over a region \(E\). Remember that when this function \(f\) is just the constant \(1\), you end up with an integral of the form \[\iiint_E dV\] and this just computes the volume of the region E! This is very useful in practice, as the volumes of complicated regions can be calculated using integration techniques.

As an example here, imagine you are an industrial engineer working for a company that manufactures sets of high-end drinkware: wine glasses, champagne flutes, and so on. The designers of the new collection have come up with a parametric model: a single model with several tune-able parameters. Their model describes the interior of the glass by a region \(E_{a,b}\), which can be tuned by changing two parameters \(a,b\): \[E_{a,b}=\{(x,y,z)\mid b(x^2+y^2)\leq z\leq a\}\]

The units here are centimeters, so the volume of the region is given in \(\mathrm{cm}^3\), better known as milliliters. The goal of this parametric is to be able to define different glass shapes by choosing different relationships between the parameters.

• Wine Glasses are shorter and fatter, so to design a wine glass we set the width and height parameters equal, \(a=b\) and get a model that depends on a single parameter.
• Champagne Flutes are taller and skinnier: to design one of these, we set the height parameter \(b\) equal to three times the width parameter, so \(b=3a\).

If the company later wants to expand its glassware collection, it does not need to go through the process of redesigning new models from scratch, it can just change the relationship between \(a\) and \(b\) to get a new look and feel.

The density of the earth varies with radius, as its composition changes from the crust to the core.

After doing a review of several cups across the industry, your manager finds the average volume of a red wine glass is 360mL. They then task you with taking the model from the design team, and constructing a wine glass as part of their collection that holds this same volume. What are the values of the parameters \(a,b\) that you need to report back to manufacturing, to have a glass of the right size?

Hint: find the volume of \(E_{a,b}\) as a function of the parameters, by doing a triple integral in cylindrical coordinates. Use the equation given to find the bounds for \(z\) and the bounds for \(r\)! Then, after finding the general formula, use the fact that this is to be a wine glass to get \(E_{a,b}\) in terms of a single parameter, and then solve the equation for this volume to equal \(360mL\) using a calculator or computer.

The Mass of the Earth

The interior structure of the Earth is very difficult to probe: we live on the very surface of the crust, a rocky layer over 20 miles thick, and to date the deepest humans have ever drilled is the Kola Superdeep Borehole, dug to a depth of 7.69 miles by the USSR (and stopped when the heat and pressure was melting through all drillbits).

Thus, it is a true wonder of modern science that we are able to calculate the density of the earth at various depths without directly retrieving samples: instead, we use the vibrations of earthquakes (and a lot of sophisticated mathematics) to deduce the density deep below the crust. To a very good approximation, the earth is spherically symmetric , and so we may record the density of the earth as a function \(D(r)\) of the *radius$ \(r\) from the center. With the density function \(D\) in hand, the mass of the earth \(E\) is given by the triple integral \[M=\iiint_E D\, dV\]

The true shape of the earth is called the Geoid, and is a slightly bumpy, slightly flattened sphere. But the total deviations from perfect spherical symmetry are less than 40km across the entire earth of diameter 12,760km: so a perfectly spherical model is more than sufficient for calculations such as mass and density!

While the true density function of the earth is rather complicated, its general behavior is well-approximated by the following function: its more dense in the center at the metal core, with density dropping off to an approximate constant level in the mantle and surface (where the earth is mostly rock, not metal):

\[D(r)=a + \frac{b}{1+r^2}\]

Where \(r\) is measured in thousands of kilometers and \(a\) is the density of rock (\(a=3,000kg/m^3\)), and \(b\) is the difference in density between rock and metal, or \(b=6,000kg/m^3\). For the coming calculations, its useful to compute these to the units of kilograms per cubic 1000km to match the units on \(r\): That is \(a=3\cdot 10^{12}\) and \(b=6\cdot 10^{12}\).

Parametric design allows multiple different objects to be expressed using the same formula, just changing the constant parameters.

Approximating the earth as a sphere of radius \(R=6,000\) kilometers, set up a triple integral in spherical coordinates that finds the mass of the earth.

Solve this integral (don’t plug in the numbers until the end, as the formula will be much cleaner if you keep \(R,a,b\) as variables), and then plug in the relevant constants. What is the mass of the earth in kilograms? (Plug the numbers in using a calculator or computer: the result will be a very large number, best reported in scientific notation!)

Sailing in the Bay

Two boats are sailing from one side of the bay to the other. In-between, the ocean water is flowing in a complicated pattern (due to tides flowing in and out of the golden gate). This water flow can be well-approximated by the vector field \(\vec{W}\) \[\vec{W}=\left\langle 2xy-y, x^2-x+1\right\rangle\]

The first boat follows the path \(\vec{r}\) and the second follows the path \(\vec{s}\) \[\vec{r}(t)=\langle t,t^2\rangle\hspace{2cm}\vec{s}(t)=\langle t^2,t^3\rangle \hspace{1cm} 0\leq t\leq 1\]

Boats sailing across the SF bay.

Compute the appropriate line integrals to answer the following question: does one boat have a more difficult journey across the bay, or is the trip equally difficult for both vessels?