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The size P of a certain insect population at time t (in days) obeys the
function:
P(t) = 500 e0.02t
Determine the number of insects in t = 0 days:
What is the growth rate of the insect population?
What is the population after 10 days?
When will the insect population reach 800?
When will the insect population double?
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Section # 4.8.
1. Uninhibited Growth or Decay:
Uninhibited Growth: N(t) = N0ekt, k>0
Uninhibited Decay: A(t) = A0ekt, k<0
N0 – initial population size
A0 – initial amount of the substance
k – growth rate (positive)
k – decay rate (negative)
t – time frame (in hours, days, months, etc.)
t – time frame (in hours, days, months, etc.)
Uninhibited growth assumes no death or bad
by-products
Half-life: a time it takes for the half of
substance to decay
Ex 1. Growth of Bacteria
The number N of bacteria present in a culture at time t (in hours) obeys the function:
P(t) = 1000e0.01t.
a) Determine the number of bacteria at t= 0
hours:
b) What is the growth rate of the bacteria?
c). What is the population after 4 hours?
d). When will the number of bacteria reach
1700?
e). When will the number of bacteria double?
f). When will the number of bacteria triple?
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Ex 2. Radioactive Decay
Iodine 131 is a radioactive material that decays according to the function:
A(t) = A0e-0.087t
Assume that a scientist has a sample of 100 grams of Iodine 131.
a). What is the decay rate of Iodine 131?
b). How much Iodine 131 is left after 9 days?
c). When will 70 grams of Iodine 131 be left?
d). What is the half-life of Iodine131?
2. Newton’s Law of Cooling:
The temperature of the heated object at a given time t can be modeled by the function:
u(t) = T + (u0 – T)ekt , k < 0
T – constant temperature of the surrounding medium
u0 – the initial temperature of the heated object
k – negative constant
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Ex 3. A thermometer reading 72⁰F is placed in a refrigerator where the temperature is a
constant 38⁰F.
a). If the thermometer reads 60⁰F after 2
b). How long will it take before the
minutes, what will it read after 7 minutes?
thermometer reads 45⁰F?
3.Logistic Growth Model:
In a logistic growth model, the population P after time t obeys the equation:
P(t) = c/(1+ae-bt),
where a, b, and c are constants with b > 0 and c > 0. The number c is called the carrying capacity
because the value the value P(t) approaches c as t approaches infinity: lim P(t) = c
t→∞
Ex 4.Suppose 6 American bald eagles are captured, transported to Montana, and set free. Based on
experience, the environmentalists expect the population to grow according to the model:
P(t) = 500/(1+83.33e-0.162t),
where P(t) is the population after t years.
a). What is the carrying capacity of the environment?
c). When will the population be 300?
b). What is the predicted population of this species of
American bald eagle in 20 years?