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Unformatted text preview: lVIATH 138 Assignment 5 (5 pages) Winter 2009 NOTE: Submit all problems marked * on Friday, February 13 1. Term Test 1 on February 23 will cover Assignments 1, 2, 3, L1, and up to #5 on Assignment 5. 11. Term Test 2 on h‘lareh 23 will cover Assignment 5 from #6  11, plus Assigmnents 6, 7, 8 and Maple Lab 1. III. Since arc length will not likely be covered until late in the week, #8 has not been assigned. It will appear on Assignment. 6. IV. Assignment 5 will require several numerical calculations — use your calculator! Note however, that on tests and exams. calculators will not be permitted, and any numeric values needed will
be given. 1. a) d) Recognizing when different physical situations give rise to the same mathematical model
is a timesaving skill. For example, consider the IVP
dy — = (10— b) dt 31(0) = yo With a > 0, b = 0, it represents exponential growth, while a < 0, b = 0 gives exponential
decay. With y = T, a, = —Ic, b = T“, and yo = To, it givos Newton’s Law of Temperature
Change. Thus knowing the solution to a single IVP gives us all three! Solve the given model and give a qualitative sketch of the solutions for each of the cases
(i) a. > 0, (ii) a < 0. You may assume b 2 0; include yo = b as one solution for each. In the rest of this problem, we explore several other possibilities. For each of them. decide
on the appropriate roles for a, b and 1 , and use your results from part The radioacative isotope C14 has a halflife of 5730 years. In 1988, the Shroud of Turin
was found to have 91% of the amount of C14 in freshly made cloth of the same material.
Does this information support or refute the claim that the Shroud was the burial cloth of
Jesus? Explain your reasoning. (Assume Jesus was buried in the year 0.) If we assume that a person’s weight depends on energy consumed (what we eat and drink)
minus energy used, one plausible model is that weight changes in proportion to the differ
ence, so dw _ = k dt where w(t) kg is weight at t days, C is the daily calorie intake, and we assume 38.5 calories
per kg per day are used. (C — 38.5w), 10(0) = wo, (i) If you wish to maintain a constant weight of 82 kg, what should be. your daily calorie
intake C? (ii) If you weigh 100 kg. and you want to lose 10 kg in a mouth, what should C be?
(Assume L: = 1.3 x 10“1 kg/calorie and a month of 30 days.) Is this result healthy"? If the purchasing power of the dollar is decreasing at an overall rate of 2% annually, how
long will it take for purchasing power to be reduced from $1 to 25 cents? MATH 138 — Winter 2000 Assignment #5 Page 2 of 6 *e) A detective finds a murder victim at 9 a.m., at which time the temperature of the body
is measured as 32.4" C. An hour later, the body temperature has dropped to 31.70 C.
Assuming the temperature of the room is a constant 200 C. use Newton’s law of cooling to
find the approximate time of death. [ HINT: Suppose t = 0 at the time of discovery, 0 am.
Note that normal body temperature is 37° C.) f) A skydiver’s velocity v obeys the DE (11.!
m— = , 0 =0
dt v() mg — k1), where m is her mass, and It is a constant “drag coefﬁcient”. *0)
*(ii) *(iii)
(iv) (V)
(vi) Find her velocity v(t) at any time t > 0. If she falls for long enough. the gravitational force mg (her weight) will be balanced
by the drag force kv. Describe what happens to her velocity once that happens, and
show how your solution in (i) predicts this. Prove that her acceleration undergoes exponential decay. (11) Thinking of your solution in as a function of mass m, find and hence show that (liri
heavier objects fall faster than light objects, according to this model [HlNTz To prove this, you will need the result 1 + z < cz for z > 0.] Does the same result hold if air resistance is ignored (i.e., \: = 0)? Explain.
. . . . . . . ch! 2 . . .
A more realistic model in this Situation 18 m] = mg — cv . If the terminal velOCity
r for a safe landing needs to be about 4 Iii/s, what should be the value of the drag
coeflicient c for a 100 kg sky—diver? [Use 9 = 9.8 iii/s2; there is no need to solve the
DE. Give the units of c as well as the numeric value] g) The von Bertlanffy growth model for the length L of a fish is (i) (ii)
(iii) (1L _
dt Given L(0) = L0 , solve for L(t), and give a physical interpretation of M. \Vhat
happens to the growth rate as the ﬁsh ages? MM — L) , where k and M are positive constants. Sketch a graph of your solution, and explain the role of It in the shape of your graph. If a certain fish is initially 6 cm long, has asymptotic length 300 cm. and reaches half
that length in 2.25 years, how long does it take to reach 90% of that length? 11) At time t = 0, you begin to learn a difﬁcult theorem. One theory of learning says that
the fraction y learned (by time t) grows at a rate proportional to the fraction remaining to
be learned. Assuming you know none of the theorem at t = 0, write down a suitable IVP for y, and use part a) to sketch a solution. What factors do you think might inﬂuence the
constant of proportionality in your model? 7in = Water flows into a reservoir of volume V = 8 x '10’3 in3 at a rate 5 x 10‘5 ni3/day, and out at the same rate. A certain pollutant has contaminated the inflow, with concentration ch, = 0.01 kg/m3. Assuming the pollutants mix uniformly
with the water in the reservoir, and there is no pollutant initially, find the quantity Q(t)
of pollutant. at time t > 0. Given that a concentration of more than 8 g/in3 is dangerous.
determine if/ when the reservoir concentration reaches this level. MATH 138  Winter 2009 Assignment #5 Page 3 of 6 j) At t = 0, the air in a meeting room with volume 500 m3 contains smoke at a concentration
of 20 parts per million (ppm). For t Z 0, fresh air enters the room at 50 m3 per minute, and
the mixture 01' smoky air exits at the same rate, while the smokers in the room continue
to add 1 ppm per minute to the smoke concentration. 1 ,
(1) Explain why (d—f‘ = 1 — 0.10 ppm/min , C(0) = 20 ppm is a suitable IVP for the concentration C(t) of smoke in the room after t minutes.
(ii) How long will it take for the concentration to be reduced to 1?. ppm? (iii) Will the concentration ever go down to 5 ppm? 2. a) At right are two solutions of the 10 N
gistic model K
dN N
—='N1——. N0=N.
dt r < K) ( ) 0 Which one has a larger value of the K intrinsic growth rate 1‘? Explain. ‘ t 1 dN 1 dy *l‘ B makin ‘ t] e 3111 ‘titution N = —. — = ——— in the 10 ‘istic model in a , solve for
)) y g l s )s y dt yz (it g )
y(t) and hence show that the solution is
7V K
N0) = N0 + (It — N0)e"‘
, d2N , K . . * . .
c) We have seen that d—fg = 0 when [\I = 7. Find the time I at winch this occurs. d) A certain population of ﬁsh follows the model in with 7‘ = ‘2 and K = 1000. Starting
at f = 0, the fish are allowed to be caught (harvested) at a rate cN. Write down a revised modeh ﬁnd the new equilibrium, and explain what happens in the two cases 0 < c < r. .. . . . (NV
(11) 0 < r < c. (No need to solve the revrsed DE; Just examine T for the two cases.)
( . *e) World population in 1990 was about 5.3 billion, with intrinsic growth rate r ~ 0.0038 per
year. Assume a carrying capacity K = 100 billion for world population. (i) Write the appropriate logistic IVP for this data, with t = 0 in 1990.
(ii) \Vrite down the solution, using the result of 21)) above. (iii) Compare the prediction of this logistic model for world population in 2005 with the
actual population in 2005. (Look it up.) 3. Common sense seems to indicate that for a population N(t) to thrive. there must be some
minimum number, say M, at any time, but that if the population gets too large, say N > K.
resources will become scarce. and it will decline. Thus a model dV
(;—t = r(N — M)(K — N), N(0) = N0. 0 < M < K . ‘r > 0, seems appropriate. Find all equilibrium solutions, and Show whether each is stable or 1101..
Then use direction ﬁeld analysis to sketch three typical solutions: 0 < N0 < M, (ii) MATH 138  Winter 2009 Assignment #5 Page 4 of 6 M < N0 < K. (iii) NO > K, and describe what happens to the population in each case. (Do
NOT solve.) d N 4. a) Show that for 0 < a < 1, the IVP dt 0 = CNI'“, N(()) = 0 has two solutions: N(t) and N(t) = (CX’CL)%. b) In a model for tumour growth, Williams and Bjerknes’r proposed that abnormal cells prop
agate according to
(IN 16
dt — 9
where time t is measured in units equal to the mean doubling time Td for normal cells.
and K > 1 is a positive constant called the ‘carcinogenic advantage’ (abnormal cells divide
K times as fast as normal cells). Given that N = 50. 000 cells are needed for a clinically
detectable tumour, use the result from a) (with suitable values of a and c) to determine
the time t at which this size is achieved for K = 1.1, and for K = 2. Convert your answers
to years by noting that Td for normal cells is 0.21333 X 10"l yr. (K — 1)N°55, (T from Models in Biology: Mathematics, Statistics, and Computing, by D. Brown and P.
Rothery) *c) For a certain breed of rabbit, the population N(t) is modeled by d N
dt (1) Show that , if there are N(()) = ‘2 rabbits initially, and 16 rabbits after 3 months, then
the population N(t) becomes unbounded at a ﬁnite time t. = CIVIDI (ii) Does the population become unbounded for any initial condition N (0) > 0, and any
c > 0‘? Explain your answer. 5. Two strains of bacteria. a mutant strain m and a normal strain it, grow exponentially according
to m(t) = mot?“ and n(t) = 7106“. However, you can only measure the fraction of bacteria of
each type, not the actual number. 77 t
a) Let p = ——1—()— be the fraction of mutants. Show that 1) satisfies the logistic growth
m(t) + n(t)
model
I) ' .
1 = A:p(1 — 7)), 19(0) = p0, Where I: = H — A and p0 = L.
(M mo + no *b) Solve the DE in a) for p(t), using separation of variables. Hence find a condition on n and
A which guarantees that the mutant strain wins out as t —> +00 (iv. p(t) —» 1). Is this
condition physically reasonable? MATH 138 — Winter 2009 Assignment #5 Page 5 of 6
_________________.————— 6. Each of the following curves represents the path of a particle moving in R2. Make a sketch of
the curve, showing the direction of motion; then ﬁnd the velocity and speed of the particle, and
locate the points (1:, y) at which the velocity is vertical, horizontal, or 0. *a) x(t) = (2 cost, 3sint), 0 S t 3 2% d) X(t) = (cos3 t,sin3 t), 0 S t S 27T
b) x(t)= <t+%,t—%>.t>0 e) x(t)=(sint,2cost). OStS27T
c) x(t) = (6"‘cost, e“sint), 0 g r. g 417 *f) x(t) = (3 sect,2tant) , mg g t g 7. For each of the following curves, find the vector equation of the tangent line at to, and state
the slope of the tangent line at to. Sketch the curve and tangent at to. [Note that in IR.2 the
slope of a vector v = (v1.1)?) is erg/UL] a) X(t) = (2t, 12), to = 1 *d) x(t) = (singt, 2cost), to = 7T/4
b) x(t) = (6“ cos t, 6" sin t), to = 7r e) x(t) = (626“), to = 0. *0 X0?) = (33,13), to = [oleI Suggestion: To see whether you’ve grasped the graphing of parametric curves, try text problem
241 on page 627. 8. Find the total distance travelled by a particle along each of the following paths. Sketch the paths. a) x(t) = (3132,19), —1 S t 31 b) x(t) = (cos3 t, sin3 t), 0 S t S 277 9. a) Compare the motion of two particles. one moving 011 the curve x(t) = (cos 7rt.sin 7rt),
O S t S 8, and the other on x(t) = (cos 7rt3, sin 7rt3), 0 S t S 2. Verify that x V X’ = 0
for both, and sketch their paths in R2. Referring to the velocity and speed of each parti
'cle, contrast their behaviour, discussing where speed x’(t) is a maximum or minimum. 1 *b) Two particles have the same speed for all t. One is moving on the path x] = (cos t. cos'2 t).
and the second is on the path x2(t) = (f(t), sin2 t). (i) What are the possible functions {(t)? (ii) If the second particle passes through (0,0) at t = 0. and (1.1) at t = g, find its path. (iii) Sketch both paths in IR.2 and compare the motion of the particles. c) Show that, for a particle moving on the path x = (R cos wt. Rsin wt), where R. and w are
2 v
constants, a(t) = E, where v = v(t) (the speed), and a(t) is acceleration.
. . V 2 2t
(1) Show that a particle movmg on the path x = t2 + 1 — 1, I. — t2 + 1 has constant speed 11. MATH 138  Winter 2009 Assignment #5 Page 6 of 6 10. Find the acceleration of a particle moving on a cycloid, with position vector x(t) = ((i(t — Sint), (1(1 — cost)). Show that the acceleration has constant magnitude. Find
the direction of motion at the top of each arch of the cycloid (t = 7T, 37T, . . Describe what
happens to the velocity at the ‘cusps’ = 2a, 471', . . (See Figure 13 on page 625 of your text.) . A particle moving in a plane has position at time t given by x(t) = (2cost,sin2t), 0 S t 3 7r Find the position and velocity of the particle at times t = 0, 1‘ = 7r/2‘ and f = 7r .
b) Find the vector equation of the tangent line to the path of the particle at to = 7r/4. c) Make a sketch of the path of the particle for 0 S t 3 7r, indicating the relevant positions
and velocities from parts a) and b) above. [ HINT: Draw the component curves, as in
Figure 8 on Text page 623.] (Optional )Find the volume of the solid formed by revolving about the xaxis the region bounded by the cycloid { a: = a (0 — 3m 6) 0 g 1: 5 27m. y=a(1—cos€) for and y = 0 [HINT: Formulate the volume integral in the usual way., and then use the parametric equations
as a substitution in the integral] ...
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This note was uploaded on 09/04/2009 for the course MATH 138 taught by Professor Anoymous during the Winter '07 term at Waterloo.
 Winter '07
 Anoymous
 Calculus

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