1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
PtichkaEL [24]
3 years ago
9

A rumor spreads through a small town. Let y(t) be the fraction of the population that has heard the rumor at time t and assume t

hat the rate at which the rumor spreads is proportional to the product of the fraction y of the population that has heard the rumor and the fraction 1−y that has not yet heard the rumor.
a. Write the differential equation satisfied by y in terms of proportionality k.
b. Find k (in units of day−1, assuming that 10% of the population knows the rumor at time t=0 and 40% knows it at time t=2 days.
c. Using the assumptions in part (b), determine when 75% of the population will know the rumor.
d. Plot the direction field for the differential equation and draw the curve that fits the solution y(0)=0.1 and y(0)=0.5.
Mathematics
1 answer:
sladkih [1.3K]3 years ago
5 0

Answer:

The answer is shown below

Step-by-step explanation:

Let y(t) be the fraction of the population that has heard the rumor at time t and assume that the rate at which the rumor spreads is proportional to the product of the fraction y of the population that has heard the rumor and the fraction 1−y that has not yet heard the rumor.

a)

\frac{dy}{dt}\ \alpha\  y(1-y)

\frac{dy}{dt}=ky(1-y)

where k is the constant of proportionality, dy/dt =  rate at which the rumor spreads

b)

\frac{dy}{dt}=ky(1-y)\\\frac{dy}{y(1-y)}=kdt\\\int\limits {\frac{dy}{y(1-y)}} \, =\int\limit {kdt}\\\int\limits {\frac{dy}{y}} +\int\limits {\frac{dy}{1-y}}  =\int\limit {kdt}\\\\ln(y)-ln(1-y)=kt+c\\ln(\frac{y}{1-y}) =kt+c\\taking \ exponential \ of\ both \ sides\\\frac{y}{1-y} =e^{kt+c}\\\frac{y}{1-y} =e^{kt}e^c\\let\ A=e^c\\\frac{y}{1-y} =Ae^{kt}\\y=(1-y)Ae^{kt}\\y=\frac{Ae^{kt}}{1+Ae^{kt}} \\at \ t=0,y=10\%\\0.1=\frac{Ae^{k*0}}{1+Ae^{k*0}} \\0.1=\frac{A}{1+A} \\A=\frac{1}{9} \\

y=\frac{\frac{1}{9} e^{kt}}{1+\frac{1}{9} e^{kt}}\\y=\frac{1}{1+9e^{-kt}}

At t = 2, y = 40% = 0.4

c) At y = 75% = 0.75

y=\frac{1}{1+9e^{-0.8959t}}\\0.75=\frac{1}{1+9e^{-0.8959t}}\\t=3.68\ days

You might be interested in
Your elevation is 2/5 of the fishes new elevation what is your elevation
Ray Of Light [21]
I need more detail like what the elevation of the fishes are

4 0
3 years ago
Joshua has more than 250 toy soldiers. When he tries to arrange them in rows of 3, there are 2 left over. When he tries to arran
Vikentia [17]
Don't look at this comment.. I messed up the first time so, yeah.
6 0
3 years ago
12. (r - 5)3 + r2 ifr= -3 and s= -4
Natasha2012 [34]

Answer:

-503

Step-by-step explanation:

the statement tell us:

(r - 5)^3 + r^2

r= -3

so we have:

(-3 - 5)^3 + (-3)^2

(-8)^3 + 9

-512+9 = -503

6 0
3 years ago
HELP!!!!!!!!!! What does each line on the protractor stand for?
Scilla [17]

Answer:

degrees like in angles. from 0 to 180

nwcggov

8 0
2 years ago
[final due to glitching] find the value of x for this parallelogram
Butoxors [25]

Answer:

x = 0

Step-by-step explanation:

both angles equal each other

I hope this helps

5 0
2 years ago
Other questions:
  • WILL GIVE BRANLIEST <br><br> Find the missing side lengths .
    14·1 answer
  • Ac set of Dishes has a list price of $399 with a 15% discount rate. What is the discount amount?
    6·2 answers
  • 1. Congruent triangles are ______ similar
    15·1 answer
  • How do I answer this question?
    7·2 answers
  • What is 3/5 of a dollar
    12·2 answers
  • Find the slope of line passing through the two points, (7, 1) and (-2, 1).
    8·1 answer
  • Round 2,182 to the nearest thousand
    10·2 answers
  • A right triangle has legs measuring 6 and 7 inches. What is the length of the hypotenuse?
    7·1 answer
  • Solve the equation -5x + 3 = 2x - 1
    15·2 answers
  • Round 45.395 to the nearest hundredth
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!