Avon High School
AP Calculus AB
Name ___________________________________________
HW 6.3B – Logistic Growth
Period _____
Score ______ / 10
1.) Because of limited food and space, a squirrel population cannot exceed 1200. The population grows at a
rate proportional both to the existing population and to the attainable additional population.
a.) Write a differential equation that describes this situation.
b.) Write the solution to this differential equation.
c.) If there were 100 squirrels two years ago and 400 one year ago, how many squirrels are there now?
(Hint: Use P(0) = 100 and P(1) = 400. You want to find P(2).
d.) Graph the first five years of squirrel
population.
You must label the tic marks of your axes with
appropriate values. Labelling every other tic mark
along the y-axis is aceptable.
e.) Use your TI-Nspire to find when the squirrel
population is growing the fastest.
2.) Zombies have attacked Avon, IN and its surrounding areas. Zombies are biting citizens of this
community and consequently are infecting normal people and immediately turning them into zombies.
(There is no incubation period – these zombies are potent!) This zombie infestation is spreading through
our population of 50,000 people at a rate proportional to both the number of people already infected and
to the number still unaffected.
a.) Write a differential equation that describes this situation.
b.) Write the solution to this differential equation.
c.) If 100 people were bitten and infected yesterday and 125 are infected today, determine how many
people will be infected and turned in zombies a week from today.
d.) Graph the first 50 days of the zombie
infestation.
You must label the tic marks of your axes with
appropriate values. Labelling every other tic mark
along the y-axis is aceptable.
e.) Use your TI-Nspire to find when the zombie
infestation is growing the fastest.
3.) Mr. Record’s “tell-all” autobiography has hit the stands and becomes quite popular. The growth of
dR
R 

readership of the book can be described by the differential equation
 4R  2 
 where R
dt
200 

represents the number of people having read the book at Avon High School and t is measured in days.
If R(0) = 3, find
a.) lim R (t )
t 
b.) the value of R that the readership is growing the fastest
4.) Ty Pist takes a typing course. He takes a pre-test and finds he can type 20 words per minute. After 1
week of the course, he can now type at 35 words per minute. The maximum typing speed of most humans
is 160 words per minute. His typing speed grows logistically.
a.) Write a differential equation that describes this situation.
b.) Write the general solution to this differential equation.
c.) Solve the differential equation for Ty.
d.) Find the rate at which the typing speed is increasing when t = 5 weeks and t = 20 weeks.
e.) Find when Ty’s growth rate is the fastest.
f.) Ty decides to quit the course when his increase in typing speed is less than 2 words per minute. What
week will Ty decide to quit?
5.) A conservation organization releases 25 Florida panthers into a game preserve. After 2 years, there are
39 panthers in the preserve. The Florida preserve has a carrying capacity of 200 panthers.
a.) Write a logistic equation that models the population of panthers in the preserve.
b.) Find the panther population after 5 years.
c.) When will the population reach 100 panthers?
d.) Write a logistical differential equation that models the growth rate of the panther population. Then
repeat part (b) using Euler’s Method with a step size of h = 1. Compare the approximation with the
exact answer.
e.) At what time is the panther population growing most rapidly? Explain.