PRE-TEST
Consider this test a
diagnostic designed to assess your ability to solve abstract mathematical
problems. You should be able to solve at
least 10 of problems 1 – 12. You ought
to be able to solve 1 of the starred (*) problems 13 - 15. If you cannot, you might have difficulty
understanding some of the concepts presented in this class. You most likely will also have difficulty
solving some of the homework problems, particularly those that require a
mathematical solution! However, I do not
require you to solve many mathematical problems on the exams in this
class. There are some mathematics-based
questions on the exams (not very many), but their degree of difficulty pales
into insignificance in comparison with those in the homework assignments. In other words, if you have trouble with
these questions, you will likely have trouble with 20% of the homework … less
trouble with the exams and overall the course will prove to be challenging for
you.
1. Write the following in scientific notation (as a power of 10):
(a) 150 thousand =
(b) 3.6 trillion =
(c) 250 million =
(d) 0.0001 =
(e) 137 thousandths =
2. Sphere A and sphere B are similar but the radius of sphere A is double the radius of sphere B.
(a) How much larger is the surface area of sphere A compared to that of sphere B?
(b) How much larger is its volume?
3. What is the value in radians of the 15 degree angle shown in the figure below?
4. Calculate the height H of the vertical line from the data given in the figure below.
5. The continents of
South America and
6. Shown in the figure below is a right circular cone. If the cone is filled with water, it holds 3.7 gallons. Suppose that all of its linear dimensions (height, radius, etc) were tripled. How much water could it then hold?
7. Shown in the
figure below is a length S of arc
that is part of a circle of radius R.
Let the ratio 
.
(a) What
is the angle 
in degrees between the
two radial lines that subtend the arc length S when k = 0.25?
(b) What is the angle in radians?
(c) Suppose k = 0.5, then what is the angle in degrees and radians?
8. Police radar guns do not measure the absolute speed of a moving car, that is, its speed relative to the ground. They measure its speed relative to the speed of the policeman’s vehicle (in which the radar gun is situated). If the policeman is not moving … for example, he is sitting under an underpass waiting to zap an unsuspecting speeder, then the radar gun does measure its speed relative to the ground … but suppose a policeman is traveling down the highway at 55 mph and spots a speeding car moving away at 75 mph one mile further down the road ahead. The policeman “hits” the moving car with his radar gun and measures his speed.
(a) What speed does the radar gun measure for the speeding car?
(b) Given the policeman’s speed, how fast does the policeman conclude that the speeding car is moving relative to the ground?
(c) If the policeman increases his own speed to 85 mph, how long will it take him to catch the speeder?
9. Most of you have been told since you were very young that the Sun is the center of the solar system and that all the planets travel in more or less circular orbits around it. Back in the 1600’s, Johannes Kepler discovered that there is a relationship between the time it takes a planet to complete an orbit (its period, T) and the average distance it is away from the Sun (its orbital radius, R). Measured in units of the Earth’s distance from the Sun, which is 1 astronomical unit, or 1 AU, and the period of the Earth’s orbit in years (which is 1 yr), this relationship is given by the equation
If Jupiter is 5.2 times further away from the Sun than is the Earth, how many years does it take Jupiter to complete one revolution about the Sun?
10. The area of a square plus two-thirds of the length of its side is numerically equal to 7/12. What is the length of the side of the square?
11. The mass of a cockroach is 1 gram. Assume that cockroaches have no predators and that their average lifespan is 2 years. Further assume that we start with a single pair of cockroaches, male and female, and that each pair produces another pair every week. How many weeks does it take before the number of cockroaches outweighs the Earth whose mass is approximately 6 x 10 27 grams?
12.* Police were summoned at midnight to a grizzly murder scene where they found the body of Eddie the Weasel, a notorious criminal with underworld connections. Upon arrival, the police officers noted that the ambient air temperature was 68 oF and the temperature of Eddie’s body was 85 oF. At 2:00 A.M., after the fingerprints were taken and suspects questioned, the body had further cooled to 74 oF.
Acting on a tip, the police apprehended Claire Voyant, Eddie’s girlfriend, passed out in her apartment. Claire had spent the evening in Louie’s Bar, drinking and threatening Eddie’s life. Eddie left and she became increasingly noisy and disruptive. Louie the bartender shut Claire off and she stormed out of the bar at 11:15 P.M. in a foul mood … probably heading for Eddie’s apartment, where presumably she murdered Eddie and then slipped out to her own apartment a few blocks away … or so the police thought! It looked like an open and shut case.
Fortunately, Claire was mathematically literate. She knew
natural logarithms and
(a) Use the formula to show that Eddie’s temperature was 85 oF and 74 oF at midnight and 2 A.M. respectively.
(b) What time was it when Eddie’s temperature was 98.6 oF? ( Could Clare have murdered Eddie the Weasel?)
13.* Prove that the
diagonal of a square is 
times the length of its side.
In other words, in the figure below, prove that 
(You cannot use the Pythagorean theorem. In essence, that’s what you’re trying to prove! You might think about trying a geometrical construction.)
14.* Archimedes was
the first mathematician to devise a method of calculating 
to any desired accuracy.
It is based on the fact that the perimeter of a regular polygon of n sides inscribed in a circle is smaller
than the circumference of the circle, whereas the perimeter of a similar polygon circumscribed about the
circle is greater than its circumference.
By making n arbitrarily large,
the two perimeters will “converge” upon the circumference of the circle, one
from above and the other from below. Or,
equivalently, the ratio of the circumference C of the circle to its diameter D
will converge on the number , i.e., 
. Archimedes started with a hexagon (see the figure below),
and progressively doubling the number of sides, stopped at a polygon of 96
sides, which yielded
or in decimal
form 
(a) Using trigonometry (which Archimedes did not and could not do), derive the following inequality relating the perimeter of the inner polygon, the circumference of the circle and the outer polygon
or if angles
are measured in radians 
where n is the number of sides of the similar polygons (take a look at the figure above, which should give you some clues).
(b) Calculate values for a 96-sided figure and show that it agrees with Archimedes’ result.
(c)
Show that as 
, the lower and upper limits both converge on .
15.* The time it takes a simple pendulum to swing once back and forth is called its period T (shown in the figure below). You might guess that the value of T should depend somehow on its mass m, its length l and the “strength,” or acceleration, of gravity g. For example, a simple experiment demonstrates that lengthening the pendulum (increasing l ) increases the period of the pendulum. If you took the pendulum to the Moon, where gravity g is weaker, and performed the experiment, the period of the pendulum would also increase, and so on. Assume that the period of the pendulum can be expressed as some combination of the above mentioned variables:
(The 
sign means
proportional to …)
a, b, and c are exponents (powers to which the variables m, l and g are raised). There is a way to calculate the values of a, b, and c using dimensional analysis. The physical properties of things have units associated with them. In the mks system, the units are meters for length, kilograms for mass and seconds for time. The units of “dynamical” things, such as a swinging pendulum, are always expressed as some combination of (i) length, (ii) mass and (iii) time, designated by the symbols [L], [M], [T]. For example, the units of speed would be [L] [T] -1. The units of the above combination of variables for the period of the pendulum must ultimately reduce to units of time [T] for the period of the pendulum. The units of m and l are [M] and [L] respectively. g is an acceleration, whose units are [L] [T] -2 … or length divided by time squared (meters per seconds squared in the mks system of units). The value of g on Earth is g = 9.8 m s -2.
Find the values of the exponents a, b, and c that give units in the above relation that reduce to units of time [T] for the period of the pendulum T.
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