Math 109 Problem of the Week
(sometimes with hints, etc.)
Last updated: 6/7/01 (Work in Progress) Watch
for the DUE DATES.
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Darts! Due June 14,
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 uniformity condition:
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.
(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?
(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.
(c) 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?
(d) 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.
2. Darts and Averages. Consider the same circular magnetic dart
board with radius 60 centimeters. Draw 5 concentric circles on the
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. [These
numbers are described as the expected score and the expected average for
the game.] Discuss your reasoning and show the work leading to your proposed
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
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:
June 21, 2001
In the sciences and engineering we measure phenomenon and use these measurements
to explain, predict, make decisions, and control what we can and cannot
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 major traffic
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 context
related to a setting connected to your major or some personal interest.
Mixing things up. Due
Suppose f is a function with its graph as in the following figure
A. Sketch graphs for the following functions on the interval
i. g(x) = f(x) + 2
ii. s(x) = f(x - 2)
iii. P(x) = 2f(x)
iv. Q(x) = f (x2)
v. R(x) = (f (x))2
B.Sketch the graphs for the derivative of f, g,
s, P, Q, and R on the interval for [-2,2].
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 .
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.
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.
A. Draw a picture that illustrates the situation as described
so far labelling points on the picture with the appropriate function values.
B. Write an equation that expresses the statement that George
always travels at the speed limit. [NOTE: The answer is not
C. 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.
Tangents to the graph of
the sine. Due
A. Prove: If a = tan(a), then the line tangent
to the graph 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
Making Curves Fit together Smoothly. Due
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.
A. 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+
and Q(x) = Dx2 + Ex+ F. Have the pair of
curves meet at (0,0).
B. Find a second pair of quadratic polynomials which can be used to
make a single smooth curve that passes through the same four points.
C. Find a single cubic polynomial that passes through the same four
D. Use some technology to sketch your solutions for A,B, and C. Discuss
briefly any noticeable differences (such as "vertices") between the three
E. [Optional- 3 bonus points for partnerships.] 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).
Some extreme problems. Due
A. 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?
B. Suppose you are given four constants A, B,
C, and D.
Value Problem related to the Tangent Function. Due
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 1000
A. Sketch a tangent (direction) field for this differential equation with
the graph (approximated) of an integral curve representing the function
Assume P(x) is a solution to the differential equation P'(x)
= 1/(1+x 2) with P(0) = 0.
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 -p /2 <
x < p /2 .
D.Use C to explain why Q(x) = x for -p /2
< x < p /2 .[ Why is Q(0)= 0?]
E. Explain why P(1) = Q(p/4) = p