Math 109 Problem of the Week
(sometimes with hints, etc.)
Last updated:3-23-2012 (Work in Progress) Watch for the DUE DATES.
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- Darts! Due Thursday 2-9-2011
1. Darts: Imagine a circular board of radius 60 centimeters
which is a powerful magnet. See Figure 1. This magnet is very
strong so that when I turn my back to it and release a magnetic
dart, the dart will be drawn to the board and will land at random
somewhere on the board. Let's suppose for simplicity the following
In other words the two events of the dart falling in two regions
of equal area are equally probable or equi-likely. Now I'll throw
a magnetic dart and mark where it lands on the board. This is our
basic experiment. For each of the following questions include a
discussion of how you arrived at your answer with your response.
- If we have any two regions on the board of equal area
then the dart would land in those regions about the same
number of times with a large number of throws.
(a) If the board is divided into two regions by a concentric
circle of radius 30 and I repeat the experiment a large number of
times, what proportion of markings from the experiment is likely
to lie in the inner region? See Figure 1.
(b) If the board is divided into four regions by concentric
circles of radius 15, 30, and 45 cm and I repeat the experiment a
large number of times, what proportion of the markings from the
experiments is likely to lie in each of the regions?
(c) What proportion of the markings from repeating the experiment
is likely for the dart to fall within A cms of the center
where 0<A<60? See Figure 2.
(d) If the board is divided into n regions by concentric circles
of radius k * 60/n cms with k = 1,2 ... n-1, and I repeat
the experiment a large number of times, what proportion of the
markings from the experiments is likely in each of the regions?
2. Darts and Averages. Consider the same circular
magnetic dart board with radius 60 centimeters. Draw 5
concentric circles on the board with radii 10,20,30,40, and 50
creating 6 regions (one disc and five bands). Consider the
following game: Throw the dart at the board and score 5 points for
landing in the inner disc, 15 points for landing in the smallest
band (between the circles of radius 10 and 20), 25 points for
landing in the next smallest band (between the circles of radius
20 and 30), 35 points for landing in the band between the
circles of radius 30 and 40, 45 points for landing in the band
between the circles of radius 40 and 50, and 55 points for landing
in the outermost band (between the circle of radius 50 and the
edge of the dart board). [In this game a low score is considered
evidence of greater skill.]
We allow a player to throw a dart 36 times and find the total
score for the player as well as the average score (the total
divided by 36).
Notice the average will be a number between 5 and 55.
Give a total score and an average that are likely for a player
with no special skill. Discuss your reasoning and show the work
leading to your proposed solution.
[These numbers are described as
the expected score and the expected average for the game.]
Suggestion: You might investigate the similar problem
with 2 regions and 4 throws and then 4 regions and 16 throws as
a way to begin thinking about the problem. How many darts do you
think would fall in each region?
Extension: (Optional) If we measure the distance each
of the 36 darts fall from the center, what do you think the
average distance would be for a player with no special
skills. Explain your reasoning, any connections with the
expected average score in the game, and whether you have any
belief about the accuracy of your response. (Is it an
underestimate or an overestimate?)
Model contexts. Due:Thursday 2-23-2012
In the sciences and engineering we measure phenomenon and use
these measurements to explain, predict, make decisions, and
control what we can and cannot measure.
For example, the context of a marsh land environment is
familiar to many residents of Arcata. Suppose that water is
flowing from a stream into a marsh for 5 hours at 30000 cubic
feet per hour. What change might be expected in the amount of
water in the marsh and in the height of the water along an
embankment at the end of the five hours. It should seem
reasonable to you that after five hours the marsh will contain
an additional 150000 cubic feet of water. We need more
information about the shape of the marsh to determine the change
in the height along the embankment. In particular we would need
more information about the surface area of the marsh and the
grade (steepness) of the embankment. We would also want to know
how fast the water is flowing out of the marsh.
Describe a context in a similar fashion related to the
following three settings. Indicate some of the important
variables and rates that would be of interest for this
context. Discuss how knowing these rates can be used to
determine other information related to the original variables.
1. Automobile and truck traffic on Highway 101 or at a
major traffic intersection.
2. A flu epidemic in a population of school children.
(Consider Susceptible, Infected, and Recovered members of the
3. A college class room. (You might consider such
variables as number of people present, temperature, or level of
Extension or Substitute for one of the above: Describe a
context related to a setting connected to your major or
some personal interest.
polynomial? Due Thursday 3-22-2012
A. Suppose f(x) = x 3 + Bx
2 +2x + A.
Find A and B so that f(0) = 1 and f '(1) = 7.
B.Let P(t) = K + Lt + Mt2 .
Find the coefficients K, L, and M so that P'(t) + P(t)
= t 2 + t + 1 for all t.
Hint: Consider the equation when t = 0, t = 1 and t = -1.
Making Curves Fit together Smoothly.
One way to make a curve that passes through several points and
looks smooth is to draw several curves that are defined by a
small number of points and make sure that when the curves are
joined together they have the same tangent lines, making the
connections appear smooth.
C. Use curves defined by two quadratic poynomial functions
to make a single smooth curve that passes through the four
points, (-1,0), (0,0), (1,-2) and (2,0) as in the figure.
Discuss briefly the strategy you used to find your solution.
Suggestion: Let P(x) = Ax2 + Bx+
C and Q(x) = Dx2 + Ex+ F.
Have the pair of curves meet at (0,0).
D. Find a second pair of quadratic polynomials which can be
used to make a single smooth curve that passes through the
same four points. [Hint: Have the pair of curves meet at
E. Find a single cubic polynomial that passes through the same
four points. [Hint: What are the linear factors of such a
A.Follow up from POW #3. Find a cubic
and a quadratic polynomial that can be used to make a single
smooth curve that passes through the five points (-2,0), (-1,0),
(0,0), (1,-2) and (2,0).
B. Fly-by-night Airlines has just announced a
special summer charter flight fare for H.S.U. students
from Arcata to Hawaii and return. A minimum of 80 students
must sign up for each flight at a round trip fare of $210.00
per person. However, the airline has offered to reduce each
student's fare by $1.00 for each additional student who joins
the flight. Under these arrangements, what number of
passengers will provide the airline with the greatest revenue
per flight? Use calculus to justify your result.What kind of
assumptions have you made about the functions you used in
solving this problem?
C. Suppose you are given four constants
A, B, C, and D.
Let g(x) = (x - A)2 + (x - B)2+
(x - C)2 + (x - D)2. Using
calculus, for what value(s) of x does g(x) assume a minimum
value? Justify your answer briefly.
Generalize your result (if possible) for
Due: Due 4-26-2012 : An Initial Value Problem related to the Tangent
Assume P(x) is a solution
to the differential equation P'(x) = 1/(1+x
2) with P(0) = 0.
A. Sketch a tangent (direction) field for this differential
equation with the graph (approximated) of an integral curve
representing the function P.
B. Estimate P(1) using Euler's method with n = 10.
C. Let Q(x) = P(tan(x)). Use the chain rule
and trigonometry to show that
Q'(x) = 1 for all x where -pi /2
< x < pi /2 .
D.Use C to explain why Q(x) = x for -pi
/2 < x < pi /2 .[ Why
is Q(0)= 0?]
E. Explain why P(1) = Q(pi/4) = pi /4.
Tangents to the graph of the sine.
A. Prove: If a = tan(a), then the line tangent
y=sin(x) at the point (a,sin(a)) passes through the origin.
[Warning:Do not assume a=0.]
B. Prove: If the line tangent to the graph of y
= sin (x) at the point (a, sin(a)) passes through the origin,
then a = tan(a).
. George and Martha's Drive. [A
Thinking Problem.] Imagine a road where the speed limit is
specified at every single point, i.e., there is a function L
where the speed limit at x kilometers from the beginning of
the road is L(x) km/hr. George and Martha are driving
their cars along this road. George's car is at a
point G(t) at time t while at the same time t Martha's
car is M(t) kilometers from the beginning of the road.
i. Draw a picture that illustrates the situation as
described so far labelling points on the picture with the
appropriate function values.
ii. Write an equation that expresses the statement that
George always travels at the speed limit. [NOTE: The
answer is not G'(t)= L(t).]
iii. Suppose that George always travels at the speed
limit and that Martha's position at time t is George's position
at time (t+5). Use the chain rule to show that Martha is always
going at the speed limit.