MATH AS A SECOND LANGUAGE

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Next: Homework Assignment for Week 2

University of Pennsylvania

PFP INTERDISCIPLINARY COURSE - SUMMER 1996

Problems from the subsection titled ``Linear Models''

1.1:
The standard price for a car to make a round trip across the Benjamin Franklin bridge (which connects Philadelphia and scenic Camden, NJ) is $2.00. But for commuters who live in Southern New Jersey and work in Philadelphia (or vice versa), it is possible to buy a special sticker which costs $25.00. With the sticker, each round trip (including the first) costs an additional $0.25.
- a.) Write down the expression which gives the total cost of making x round trips across the bridge at the regular $2 rate. In Maple, define the variable r to be equal to this expression (Don't forget to use the asterisk for multiplication!).
- b.) Write down the expression which gives the total cost of making x round trips across the bridge for a commuter with a sticker. In Maple, define the variable c to be equal to this expression.
- c.) Use Maple to plot r and c for x varying from 0 to 30 on the same set of axes. The easiest way to do this is to type
```
                      plot({r,c}, x=0..30);
```
  Describe in words what you see and what it means for the typical commuter.
- d.) There is a value of x (not an integer) for which r and c are equal. You can find it with Maple using the statement
```
                      solve(r=c, x);
```
- e.) So, how many trips does a commuter have to make in order for it to be worth her while to buy a sticker? Given that a sticker is valid for one month, should the typical commuter buy one? Why or why not?
1.2:
Do a five-step analysis as in the previous problem for the following situation: You are about to buy your first (used) car. You have a choice of buying a 1985 Chevy Caprice for $3200 or a 1985 Honda Accord for $4400. Your mechanic has assured you that both cars are in good shape and should give you many repair-free miles (you are quite a lucky consumer). The Chevy gets 16 miles per gallon and the Honda gets 35 miles per gallon. Assume that gasoline costs $1.25 per gallon. How many miles would you have to drive before the cost of buying and fueling the Chevy becomes greater than the cost of buying and fueling the Honda?
1.3:
You know that the equation of a line is y = mx + b. To get some practice applying Maple to try to understand how varying the parameters of a family of curves affects the curves, we will see how to do it for lines.
- a.) In Maple define a variable n to be the expression 3*x-5. We will use this n as the first item in a list of expressions to be plotted later. To continue the list, we can use an iterative statement as follows:
```
               for b from -4 to 5 do n:=l,3*x+b; od;
```
  When you ask Maple for the value of n after executing this statement, it will report that n is the list of expressions 3x-5, 3x-4,..., 3x+4, 3x+5. We can plot all of these expressions on the same set of axes using a statement like:
```
                plot({n}, x=-3..3);
```
  This should illustrate clearly the effect of changing b in the equation of a line (but you already knew that, didn't you?)
- b.) Generate and plot a list of linear expressions which illustrate the effect of changing m in mx + b.
Problems from the section named ``Quadratic and Polynomial Models"
1.4:
Generate and graph three lists of quadratic functions (as we did in exercise 1.3 above) which illustrate the effects of changing a, b, and c in a quadratic expression such as y = ax² + bx + c. Write down in words what you observe about the various effects.
1.5:
Use Maple to plot the graphs of x, x² , x³ , ..., x¹⁰ either all on the same axes or else in different windows in such a way that you can have several of them on the screen at once. Explain in words how the shape of the graphs for even powers of x\ differ from the shape for odd powers.
1.6:
The most general cubic expression is y = ax³ + bx² + cx + d. Use Maple to investigate how the shape of the graph of the cubic depends on a, b, c and d. Also examine the effect of plotting your cubic over larger and larger ranges (such as {x=-10..10}, then { x=-50..50}, etc.). What can you say about the ``global behavior'' of all cubics? Be complete in your description! The brevity you normally use for answering technical questions is NOT proper for this course.

Problems from the section -- One Dimensional Motion
1.7:
The graph below shows velocity vs. time for a remotely controlled toy car moving along a floor. Find the following:
- a.) The average acceleration for the time between 0 and 8 seconds.
- b.) At what time (or times) does the acceleration of the toy car have its greatest positive value? What is the acceleration at those time(s)?
- c.) What is the average acceleration for the time period between 2 and 4 seconds?
1.8:
The graph below is supposed to represent the position versus time for a cyclist. On the basis of this graph, do the following:
- a.) Describe in words and numbers the motion of the cyclist as depicted by this graph. You should use terms like speed, acceleration, position, etc. and should be sure to include numerical values for these, in other words, make a precise description of the cyclist's motion. Is there anything peculiar about any part of the motion?
- b.) Now describe the cyclist's motion with equations, e.g. write down the equation that describes the motion from 0 to 5 seconds, the equation for 5 to 10 seconds, etc. It's a good idea for you to check your equations by plotting them with Maple and comparing to the original.
- c.)Find the average acceleration and average velocity for the time between 0 and 20 seconds.
1.9:
A basketball, initially at rest, is dropped from the top of a very tall building by an intrepid and unwise student of physics. Neglecting air resistance,
- a.) Calculate the position and velocity of the ball after 1 second, 2 seconds, and 3 seconds.
- b.) Make a plot, using Maple, of the velocity of the ball for times between 0 and 5 seconds.
1.10:
A baseball player hits the ball straight up from home plate. The ball remains in the air for 12 seconds before being caught by the catcher. How high did the ball go?
1.11:
A golfball is thrown straight up from the top of a building 50 meters high. The initial speed of the golfball is 20 m/s and it just misses the edge of the roof when it eventually falls. Neglecting air resistance, find
- a.) The time needed for the golfball to reach its maximum height.
- b.) The time needed for the golfball to return to the height from which it was initially thrown
- c.) The velocity of the stone just at the instant it hits the ground.
1.12:
A "superball" is dropped from a height of 2 meters above the ground. After hitting the ground, the ball bounces to a height of 1.85 meters, where it is caught. Find
- a.) What is the velocity of the ball just after it makes contact with the ground?
- b.) Ignoring the time the ball is in contact with the ground, find the total time required for the ball to fall from its dropping point and travel to the point where it is caught.
1.13:
A basketball, initially at rest, is dropped from the top of a very tall building by an intrepid and unwise student of mathematics. Whenever the basketball hits the pavement at the bottom of the building, it bounces straight up, leaving the pavement with a speed which is 7/10 's the speed with which it struck the pavement. If the height of the building is given as a known quantity which we will call h, find the total distance traveled by the ball in terms of h. Neglect any effects due to air resistance.
See the action here!
1.14:
An interesting problem for non-constant acceleration: In 1979, one of the first magnetically levitated trains was tested in Japan. The mag-lev train is suspended and propelled by magnetic forces. The nature of these forces provides the train with a constant acceleration and the same magnitude constant deceleration. So, throughout the first half of a typical mag-lev journey, the train is accelerating and throughout the second half the train is decelerating. The test train traversed a straight 7000-meter-long track, starting from rest, and it reached a maximum speed (at the midpoint of the journey) of 144 meters per second. The problem is to find the value of a, the constant acceleration, and how long the trip took.
1.15:
A car is traveling along a straight tollway in New Jersey. Initially, the car is stopped at a toll booth. At time t = 0, the car pulls away from the toll booth with a constant acceleration for 10 seconds. 10 seconds after leaving the toll booth, the car is moving at a speed of 100 feet/second. 100 seconds after leaving the first toll booth, the car begins to decelerate as it approaches a second toll booth. 20 seconds later, the car is stopped at the second toll booth.
- a.) Make plots of the car's velocity and acceleration as functions of time from 0 to 120 seconds.
- b.) The speed limit in New Jersey is 55 miles/hour. Did the driver of the car break the speed limit?
- c.) What is the distance between the two toll booths?
1.16:
A jetliner needs to achieve a speed of 100 m/s for takeoff. The acceleration of the jetliner is 3 m/s² . The brakes for the jetliner provide a maximum deceleration of 1 m/s² .
- a.) If are you asked to design a runway for an airport for such a jetliner, what must the minimum length of the runway be for takeoff?
- b.) In actuality, the design of the runway should include some extra room for an emergency, such as an aborted takeoff. Given a runway length of 3 km and the jet parameters above, how long does the pilot have to decide whether or not to abort if the plane is not to run off the end of the runway? Click here for a pictorial hint.
1.17:
The engineer on a commuter train can minimize the time t between two stations by accelerating (a₁ = 0.1 m/s² ) for a time t₁, then causing a negative acceleration (a₂ = -0.5 m/s² ) by using the brakes for a time t₂. Since the stations are only 1 km apart, the train never reaches its maximum velocity. Find the minimum time of travel t, and the time t₁.
1.18:
The diagram and video above show the passing ability of the automobile at low speed. From the data supplied in the figure, calculate the acceleration of the automobile during the pass and the time required for the pass. Assume constant acceleration. After finding the acceleration and the time required for the pass, use Maple to plot the position of both the automobile and the truck as a function of time. Using the graph, find the time corresponding to the point at which the automobile just passes the truck (i.e. the back of the car just passes the front of the truck).
(To see a visual hint here, click on the picture for this problem.)
1.19:
An astronaut needs to retrieve an errant satellite located 50 m from the astronaut's space shuttle. The astronaut dons an MMU (manned maneuvering unit), exits the shuttle, and accelerates towards the satellite at a rate of 0.2 m/s² for 10 seconds. After the 10 seconds, the astronaut cuts the thrusters and drifts towards the satellite at a constant velocity until making contact with it. Determine how long it takes for the astronaut to reach the satellite and the distance of the astronaut and satellite from the space shuttle at the time of contact.

Problems from subsection "Functions Useful for Science"
1.20:
Use Maple to plot several exponential functions on the same set of coordinate axes. Write in words what the role of a and b are. Try to use words that have some correspondence with the ideas of slope and y-intercept of lines.
1.21:
Verify the statement that the quantity is independent of which two points on the graph of y = ab^x are chosen. What is the common value of the quantity?
1.22:
Verify that the ratio of populations in successive years in Mexico is essentially constant (about 1.026). Write an exponential function of the form P = ab^t which approximates the data in the chart, where t is the number of years after 1980 (you must choose a and b). Use Maple to make a table of both the data and the values of your function. For example, you might define a list with the data in it as follows: l:= [67380000,69130000,70930000,72770000,74660000,76600000,78590000]; and then, once you have defined your function to be f(t), say, you can make the table via the command:
```
       
             for x from 0 to 6 do print(x,l[x+1], f(x)); od;
 
```
How far off is your function from the actual data?
Find the doubling time for the population of Mexico using the equation you found above.
1.23:
For the data given on kerosene purification, assume P₀ = 1 and make a table of and plot P(n) for various n using Maple. How many feet of pipe does it take before only half of the pollutants remain? You can estimate this from your table and graph, but how do you find out exactly? How many feet before only 10% of the pollutants remain?
1.24:
Find the formula for half-life in terms of b. Also find the formula to write down equations which model the decay of Carbon-14 (half-life 5730 years), Uranium-235 (half-life 713 million years) and Uranium-238 (half-life 4.51 billion years).

Comparisons and combinations of exponentials,polynomials, and trigonometric functions
1.25:
Let's consider two functions: f(x) = x¹⁰ and g(x) = 2^x. Use Maple to plot the two functions on the same axes for {x=0..5}. Which one is bigger? How about for { x=0..15}? { x=0..50}? Which function wins out ultimately?
1.26:
A good test of which of two functions that are both increasing is increasing the fastest is to consider the quotient of the two functions. Use Maple to explore what happens to the quotient f(x)/g(x) as x gets bigger and bigger. Do this both graphically and numerically (remember to use values of x with decimal points so that Maple doesn't try to evaluate 2 to the x exactly!).
There is an important lesson here -- any increasing exponential function eventually increases faster than any polynomial, and the product of any decreasing exponential function with a polynomial will approach zero as x gets larger and larger.
1.27:
Free play -- experiment with various sums, differences, products, and quotients of polynomials and exponentials. Choose at least 6 such combinations. For each of your choices, first try to predict approximately what the graph will look like, both for values of x near zero and also for extremely large (and negatively large) values of x. Then use Maple to plot graphs of your functions for various (large and small) ranges of x. Try to produce interesting-looking graphs, and also try to develop a feel for which combinations for these functions produce which kinds of graphs.

Problems from the section titled ``Trigonometric Equations''
1.28:
Everybody remembers the standard ``sine wave". Have Maple plot the functions sin(x), sin(2x), sin(4x), 2 sin(x) and a few others of your own choosing for values of x between 0 and 4 time pi. Also try combinations like 2 + sin(x), sin(x) + 2 cos(x), etc. Using words like ``period'', ``amplitude'', and ``frequency'', write verbal descriptions of your graphs.
1.29:
More fun -- Now combine sines and cosines with polynomials and exponential functions via multiplication (for example 2^xsin(6x) is fun) and addition. Use your intuition as you did in exercise 1-18 to produce interesting graphs.
Composition of functions is an especially useful way to produce new functions from old ones. In the process of function composition, you place the expression for one function in the parentheses of another. For example, if f(x) = 2^x and g(x) = -x², then f(g(x)) = 2^-x². Maple is quite good at calculating the results of function composition.
1.30:
Use Maple to verify the preceding statements. There are two ways to do this. The first is to define f and g as functions via the statements
```
                       f := x -> 2^x: and g := x -> -x^2;.
  
```
Then type
```
                      f(g(x));
  
```
to see the composition. The other way is to define f and g as Maple expressions, via
```
                     f := 2^x; and g := -x^2;
  
```
and then do the composition with the statement
```
                    subs(x=g,f);
  
```
(which substitutes g for x in the expression for f). Use either method and graph the result. Does it look familiar?
1.31:
Choose several different pairs of functions f and g. For each, use Maple to compute f(g(x)) and g(f(x)). For each composition, use Maple to draw the graphs of f and g first, then try to imagine what the graph of the two compositions should look like. Then verify your intuition by using Maple to graph the two compositions. Do you think you could find two ``really different'' functions f and g such that f(g(x)) = g(f(x))?

Problems from the section titled ``Parametric Equations''
1.32:
- a.) Use Maple to make a table of positions of the ball as the time t varies from 0 to 0.5 seconds. Use a statement such as
```
       for t from 0 by 0.05 to 0.5 do print(t, [x,y]); od;
  
```
  after defining x and y appropriately. Note that Maple uses square brackets rather than our usual parentheses when dealing with ordered pairs.
- b.) Use Maple to fsolve for the time when the ball hits the ground (i.e. when y=0). Be sure to ``undefine'' the variable t (using t:='t') before trying to solve. You will get two roots. Use b := [2] to assign the positive one to the variable b.
- c.) Maple understands parametric equations -- ask it to
```
                plot[x,y,t=0..b]);
  
```
  to get a picture of the ball's trajectory (note that the range of the time variable must be given as an extra element of the ``ordered pair''). Is the graph what you expected?
1.33:
- a.) Use Maple to plot several examples of circles (maybe even families of circles) with different radii and centers.
- b.) An important problem in kinematics is to understand the motion of a point on the rim of a wheel. Suppose the wheel has radius a, and is rolling from left to right along the x-axis at a speed of a units/second (thus, the angular velocity of the points on the rim of the wheel is 1 radian/second with respect to the center of the wheel). Then a point on the rim of the wheel is going around a circle, as the center of the circle is itself moving along the line y = a. In a few complete sentences, explain why the parametric equations of the path of a point on the rim of the wheel are
```
                     x = at + a cos(t) 
                     y = a + a sin(t)
  
```
- c.) Use Maple to plot this curve. It is called a cycloid and arises in a surprising number of different places in mathematics and physics. If you are using Maple V, Release 2 (on a Windows machine, for example), have Maple do an animation which shows the circle moving from left to right, and the path of the particle as it moves. (Hint: To get the correct correspondence between the moving circle and the cycloid curve tracing the point's path, you may find it useful to introduce a change in phase to your cycloid equations so that
```
                      x = at - a sin(t) 
                      y = a -  a cos(t)
  
```
  Remember that the phase can turn a sine into a cosine and vice versa since sin(Pi/2 - theta) = cos(theta)
- d.) Can you figure out how the equations change if the wheel is rolling uphill -- say, along the line y = x/2? Find the parametric equations for the path of a point on the wheel in this case.
- e.) What if the wheel is rolling around the outside of another (circular) wheel (think of gears)? Around the inside? Use the unit circle as the fixed wheel and roll your wheel of radius a round its outside and inside. You could get carried away, and try to figure out equations for a point on the rim of a wheel that is rolling around the inside of another wheel which in turn is rolling along a line, etc ...
1.34:
Plot several examples of Lissajous figures. You may have to supply the option
```
              numpoints=100
  
```
to get nice smooth curves. This option goes after a comma between the right bracket and right parenthesis of the plot function. Its purpose is to force Maple to compute more points than it usually does before it tries to connect them with an interpolating curve.
1.35:
More free play -- use different combinations of functions in parametric equations to try to generate interesting-looking curves. Maybe we'll have an ``art show'' of the most striking examples you come up with. Here's two to get you going:
```
  plot([sin(4*t)*cos(t), sin(4*t)*sin(t), t=0..2*Pi], numpoints=100);
  
```
```
  f:=x->exp(-(x-3)^2*cos(4*(x-3)));
  plot3d([r,f(r)*cos(t),f(r)*sin(t)],r=1..5,t=0..2*Pi,grid=[35,25],axes=FRAME);
  
```
Problems from the subsection titled ``Classification of Matter''
1.36:
From our "everyday" experiences we realize that air expands on heating. Thus, the density of the air (remember density = mass/volume) should decrease as it is heated. It is for this reason that balloons rise when inflated with warm air. About a hundred years after Boyle's J-tube experiments, an amateur balloonist in France, by the name of Jacques Charles (1746-1823) - made a series of careful investigations of the effect of changing temperature on the volume of a sample of air. Specifically, he investigated the change of volume of (a fixed amount of) air as a function of temperature - at fixed pressure. A possible set of data is recorded below. The temperature is given in degrees Celsius (° C) and the volume is given in cubic centimeters (cm³ ).

Temperature (° C) Volume (cm³ )

10.0 141

20.0 145

25.0 148

30.0 150

35.0 153

50.0 161

80.0 175
- a.) Enter the above data in MAPLE as an ordered list of ordered pairs (x,y) - where temperature is the independent variable ("x") and volume is the dependent variable ("y"). Remember to use brackets - not braces or parentheses, i.e. datalistname := [[t₁, V₁], [t₂, V₂], ...].
- b.) Using MAPLE, plot volume (in cm³ ) as a function of temperature (in ° C) and obtain a point-plot (style = POINT). You might want to change the style of points to something bigger than just a literal point so that you can clearly see the data.
- c.) Next, use MAPLE to determine the best-fit line for the data plotted in (a). The equation should be of the form: V = at + b where V, t, a, and b are the volume (in cm³ ), the temperature (in ° C), the slope, and the y-intercept, respectively, of the best-fit line. You must load MAPLE's statistics package (use the with(stats) command). Before you run the least-squares fit procedure you must convert your data to two arrays, one for x data and one for y data (i.e. xarrayname := [x1, x2, x3, ...], yarrayname := [y1, y2, y3, ...]). Make the fit to a best fit line using Maple's fit[leastsquares...] command. List the final equation for the best fit line and give the slope and y-intercept in appropriate units - according to the units of your input data.
- d.) Display on the same graph, the plot of the best-fit line and the point-plot from (b). Remember to load the display procedure, i.e. with(plots,display). The form of the display command is display({firstplotname,secondplotname,...}). In order to run the display in the form shown you must have already performed each of the plots (even if they have not been previously displayed) and named them so that these names are used in the display statement (i.e. firstplotname and secondplotname in the above example).
- e.) Often we use the words interpolate and extrapolate when referring to the application of an empirically derived mathematical relationship to data between (interpolate) measured points and outside (extrapolate) of measured points. By suitable extrapolation of your best-fit line, determine the temperature (in ° C) for which the air sample would have zero volume. This temperature was termed the "absolute" lowest temperature attainable - since it was realized that if a substance could reach a temperature lower than this value, it would have negative volume - an obvious absurdity!
- f.) Reformulate your best-fit line equation in terms of a new temperature scale such that the zero volume corresponds to a temperature "T" of zero. Define this new temperature scale (T) in terms of Celsius scale temperature (t) and suitable data from your best-fit line. This reformulated best-fit line should be designed to have the same slope as its predecessor. Then, write down this reformulated equation: T = a't + b' and specify a' and b' in appropriate units.

Temperature (° C)	Volume (cm³ )
10.0	141
20.0	145
25.0	148
30.0	150
35.0	153
50.0	161
80.0	175

1.37:
Below is some of Boyle's original data from the "J-tube" experiment - slightly modified. In Boyle's original experiment, he measured the pressure head of mercury (Hg) as a function of the length of the air column. The following data are presumed to be recorded at a constant temperature for the fixed amount of air.

Length of gas column (l)	Pressure head (h)
in inches	in inches of Hg
19.7	0.0
17.7	3.4
15.7	7.4
13.8	12.7
11.8	19.8
9.8	29.8

By referring to the apparatus design of the J-tube and using the added piece of information that the barometric pressure during these measurements was 75.0 cm of Hg, do the following. You also need to know the cross-sectional area of the J-tube in the region of the trapped gas. You may assume that the cross-sectional area is uniform and equal to 0.25 square inches.

a.) Determine the pressure (P) of the air - in atmospheres - for each trial.
b.) Determine the volume (V) of the air - in liters - for each trial.
c.) Determine - using MAPLE - what (if any) of the following relationships will best approximate a line. You may try relationships of the form: P = aV + b, P = aV² + c, P = a/V + b, P = a/V² + b, using MAPLE's statistical regression package (see problem 1-27). When a suitable form is found, determine the slope and y-intercept of the corresponding line, in appropriate units.
d.) Display on the same graph, the plot of the best-fit line and the point-plot of the data in the appropriate form.
e.) Using your best-fit line (equation or plot), determine one interpolated pressure-volume coordinate pair and one extrapolated pressure-volume coordinate pair.
f.) According to the Register Book of the Royal Society dated September 11, 1161, an excerpt of Boyle's account and the assertions are as follows:
"...all that I shall now urge being, that however, the tryal already made sufficiently proves the main thing for which I here alledge it; since by it 'tis evident, that as common air when reduc'd to half its wonted extent, obtained near about twice as forcible a spring as it had before: so this thus-compressed air being further thrust into half this narrow room, obtained thereby a spring about as strong again as that it last had, and consequently four times as strong as that of the common air."
What is the quantitative relationship between pressure and volume that Boyle is suggesting? Does this agree with what you found in (c)? Carefully explain why or why not.
1.38:
Referring to the reading as needed, begin with the ideal gas law (PV = nRT) and derive the mathematical expression for the relationship between density (d) and molecular weight (M) of a gas that obeys the ideal gas law. For the data below - obtained for a fixed amount of a certain gas at a constant temperature of 25° - suitably process this data so that your derived expression above can appropriately be applied to determine the molecular weight of this gas. Express your answer in grams/mole. Recall that "real" gases exhibit truly "ideal" behavior only in the limit of very low pressures. Hint: "extrapolate".

Pressure (torr) 91.74 188.98 277.3 452.8 639.3 760.1

density (g/L) 0.225 0.456 0.664 1.062 1.468 1.734

1.39:
Consider the following compounds (I-VIII) and the recorded data below. The elemental compositions are listed below in grams. The elements represented are the "mystery elements" X, Y, Z. Data are also recorded for H₂(g) and for O₂(g). All densities are recorded for gaseous compounds at 25° and 1 atm. You will use the Cannizzaro method (discussed in class) to determine the following for each compound. Maple has a Greatest Common Factor procedure - so you may use it if you want to. Remember, use of the Cannizzaro method does not make explicit use of the ideal gas law!! Refer to the reading as needed.

a.) At the given conditions, determine the average value of "k" (in L/mole) for the proportionality between the gas density (d) in g/L and the molecular weight (M) in g/mole.
b.) Determine the molecular weight (M) in g/mole of each compound.
c.) Determine the (mass) percent composition of each element in each compound.
d.) Determine the total mass of each element per mole of compound (in grams element/mole of compound).
e.) Determine the most likely atomic weight of each element (in g/mole) - using the Greatest Common Factor (GCF) Method.
f.) Determine the probable molecular formula for each compound - in terms of X, Y, and Z
g.) H₂ (gas) density at given conditions = 0.08175 g/L

h.) O₂ (gas) density at given conditions = 1.3079 g/L

Elemental analysis of sample - all masses in grams (g)
Compound	d (g/l)	Mass of sample	X	Y	Z
I	4.210	0.8500	0.2641	0.5859	---
II	5.518	0.7000	0.3318	0.3682	---
III	4.414	0.7500	0.2222	---	0.5278
IV	7.112	1.2500	0.2299	1.0201	---
V	5.968	1.4500	0.3178	---	1.1322
VI	4.169	1.1500	0.7216	---	0.4284
VII	2.228	1.3000	---	0.8468	0.4532
VIII	3.781	1.2000	---	0.4605	0.7395

Extra Credit: Identify elements X, Y, and Z. You'll have to refer to a periodic table.

1.40:
Avogadro revisted. A gaseous compound known to contain only C, H, and N is mixed with exactly the volume of O₂(g) needed for complete combustion to CO₂, H₂O, and N₂ - all gases. At STP (1 atm and 273 K), burning 9 volume of mixture (compound + O₂) produces 4 volumes of CO₂(g), 6 volumes of H₂O(g), and 2 volumes of N₂(g).
- How many volumes of O₂(g) are required for complete combustion?
- What is the molecular formula of the compound?

Lab I: Finding the Measure of Things

Most of our knowledge of how the world works comes from application of the scientific method. We observe things about which we are curious, make hypotheses about what's going on, develop hypotheses into theories by using mathematics to make them quantitative, then experimentally test the predictions of the theory. Through the scientific method, human intuition and curiosity have extended our understanding into realms as small as quarks and as large as the universe itself. The first part of the method is observation. To be useful, our observations must be detailed and accurate and our description of the details must be clear, concise, and quantitative. We need to measure things in terms of commonly agreed upon standards and describe the measurements so that others who are curious to understand the observations can repeat them in an unambiguous way.

For this experimental exercise, you are asked to practice your observation skills by measuring large objects that are found on campus and then providing a detailed description of how you did the measurement and what results you got. Not every group will have the same object to measure, but enough groups will to allow for comparison of methods based on creativity and accuracy. Each group should do its OWN measurement! There are many ways to perform observations. The importance in science of comparing independent methods can not be stressed enough.

The objectives for each group are listed below. Unlike most science laboratory exercises with which you may be familiar, this exercise will provide no equipment and no guidelines as to how to do the measurement. All we ask is that you exercise common sense in terms of safety. Don't do anything which is dangerous! In particular, climbing to the top of tall objects or getting onto the roof of a building without prior permission are not allowed.

Your write-up should be at least 1½ pages in length and should thoroughly discuss your method of measurement (or measurements) including a listing of any equipment you may have used, the theory behind your measurement technique, and the accuracy of your measurement. The last point, accuracy, is very important since some measurement methods are more accurate than others, hence any valid quantitative measurement should include an estimate of its accuracy for comparison with other measurements. Feel free to include any drawings or Maple plots which illustrate your technique or results. In addition, graphs are certainly appropriate (in fact necessary) in most cases in which an average measurement is used. The write-up should also include a description of where and when you made your observations (you can get different numbers depending on where on the object you did your measurement). Again, we are stressing completeness in your written description. If you have trouble getting started, ask your instructors or peer advisors for suggestions. Your objectives are below.