Math 109 Problem of the Week
(sometimes with hints, etc.)
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- Darts! Due Thursday 2-3-2011
1. Darts: Imagine a circular board of radius 60 centimeters
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.
suppose for simplicity the following uniformity condition:
In other words the two events of the dart falling in two regions of
area are equally probable or equi-likely. Now I'll throw a magnetic
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
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
circle of radius 30 and I repeat the experiment a large number of
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
lie in each of the regions?
(c) What proportion of the markings from repeating
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
by concentric circles of radius k * 60/n cms with k = 1,2 ...
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
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
bands). Consider the following game: Throw the dart at the board and
5 points for landing in the inner disc, 15 points for landing in the
band (between the circles of radius 10 and 20), 25 points for landing
the next smallest band (between the circles of radius 20 and 30), 35
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
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
and an average that are likely for a player with no special skill.
Discuss your reasoning and show the work leading to your
numbers are described as the expected
score and the expected average
Suggestion: You might investigate the similar problem with
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
Extension: (Optional) If we measure the distance each of
36 darts fall from the center, what do you think the average distance
be for a player with no special skills. Explain your reasoning,
connections with the expected average score in the game, and whether
have any belief about the accuracy of your response. (Is it an
or an overestimate?)
In the sciences and engineering we measure phenomenon and use these
to explain, predict, make decisions, and control what we can and cannot
For example, the context of a marsh land environment is
to many residents of Arcata. Suppose that water is flowing from a
into a marsh for 5 hours at 30000 cubic feet per hour. What change
be expected in the amount of water in the marsh and in the height of
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
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.
particular we would need more information about the surface area of the
marsh and the grade (steepness) of the embankment. We would also want
know how fast the water is flowing out of the marsh.
Describe a context in a similar fashion related to the
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
1. Automobile and truck traffic on Highway 101 or at a
2. A flu epidemic in a population of school children. (Consider
Susceptible, Infected, and Recovered members of the population.)
3. A college class room. (You might consider such variables
as number of people present, temperature, or level of sound.)
Extension or Substitute for one of the above: Describe a
related to a setting connected to your major or some personal
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.
Curves Fit together
One way to make a curve that passes through several points and looks
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
tangent lines, making the connections appear smooth.
C. Use curves defined by two quadratic poynomial functions to
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
to find your solution.
Suggestion: Let P(x) = Ax2 + Bx+ C
and Q(x) = Dx2 + Ex+ F. Have the pair
curves meet at (0,0).
D. Find a second pair of quadratic polynomials which can be used
make a single smooth curve that passes through the same four points.
[Hint: Have the pair of curves meet at (1,-2).]
E. Find a single cubic polynomial that passes through the same
points. [Hint: What are the linear factors of such a polynomial?]
Due Thursday 3-24 A.Follow up from Problem #3. Find a cubic and
polynomial that can be used to make a single smooth curve that passes
the five points (-2,0), (-1,0), (0,0), (1,-2) and (2,0).
B. George and Martha's Drive. [A
Imagine a road where the speed limit is specified at every single
i.e., there is a function L where the speed limit at x kilometers
the beginning of the road is L(x) km/hr. George and Martha are
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)
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
ii. Write an equation that expresses the statement that George
always travels at the speed limit. [NOTE: The answer is not
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).
the chain rule to show that Martha is always going at the speed limit.
Due Thursday 4-7: Some extreme
problems. A. Fly-by-night Airlines has
just announced a
summer charter flight fare for H.S.U. students from Arcata to
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
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?
B. Suppose you are given four constants A,
C, and D.
Let g(x) = (x - A)2 + (x - B)2+
C)2 + (x - D)2. Using calculus, for what
of x does g(x) assume a minimum value? Justify your answer briefly.
Generalize your result (if possible) for 1000
Due: Thursday 4-28: An
Value Problem related to the Tangent Function.
Assume P(x) is a solution to
differential equation P'(x)
1/(1+x 2) with P(0) = 0.
A. Sketch a tangent (direction) field for this differential equation
the graph (approximated) of an integral curve representing the function
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
pi /2 .[ Why is Q(0)= 0?]
E. Explain why P(1) = Q(pi/4) = pi
A. Prove: If a = tan(a), then the line tangent
of 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