EXPONENTIAL GROWTH AND DECAY
217
26.20
26.21
26.22
26.23
26.24
26.25
26.26
26.27
26.28
26.29
If y is the number of flies y = 20e
005 '. When the enclosure is full, y = 640. Hence, 640 = 20e
005 ',
32 =e
005 ', 0.05f = In 32 = In (2
s ) = 5 In 2 = 3.4655. Hence, r = 69.31. Thus, it will take a little more than 69
days to fill the enclosure.
The number of bacteria in a certain culture is growing exponentially. If 100 bacteria are present initially and 400
are present after 1 hour, how many bacteria are present after 31 hours?
The growth equation is y = 100e
Kt . The given information tells us that 400 = 100e
K
, 4 = e
K . At
t = 3.5, y = !OOe
3iK = 100(e*)
3 5 = 100(4)
7
'
2 = 100(2)
7 = 100(128) = 12,800.
A certain radioactive substance has a half-life of 3 years. If 10 grams are present initially, how much of the
substance remains after 9 years?
Since 9 years is three half-lifes, ( \ )
3
10 = 1.25 grams will remain.
If y represents the amount by which the temperature of a body exceeds that of the surrounding air, then the rate at
which y decreases is proportional to y (Newton's law of cooling). Ify was initially 8 degrees and was 7 degrees
after 1 minute, what will it be after 2 minutes?
Since D,y = Ky, y = y 0 e
K> = 8e
Kl . The given facts tell us that 7 = 8e
K , e
K =\. When f = 2, y =
8e
2K = 8(e
K )
2 = 8(I)
2 = 6.125 degrees.
When a condenser is discharging electricity, the rate at which the voltage V decreases is proportional to V. If the
decay constant is K = -0.025, per second, how long does it take before V has decreased to one-quarter of its
initial value?
V=V 0 e~°025 '. When V is one-quarter of its initial value, \V 0 = V 0 e'
0 '
025 ', \=e~°025 ', -0.025f =
In J = -In 4 =-2 In 2 = -1.3862. Hence, f = 55.448.
The mass y of a growing substance is 7(5)' grams after t minutes. Find the initial quantity and the growth constant
K.
y = 7(5)' = l(e
1 "
5 )' = 7e
(ln 5)
'. When t = 0, y = 7 grams is the initial quantity. The growth constant K
is In 5.
If the population of Latin America has a doubling time of 27 years, by what percent does it grow per year?
The population y = y 0 e
Kl . By the given information, 2y 0 = y 0 e
27K , 2=e
27A: , e
K = V2« 1.0234. By
the solution to Problem 26.6, the percentage increase per year r = 100(e* -1) = 100(1.0234 - 1) = 2.34.
If in 1980 the population of the United States was 225 million and increasing exponentially with a growth constant
of 0.007, and the population of Mexico was 62 million and increasing exponentially with a growth constant of
0.024, when will the two populations be equal if they continue to grow at the same rate?
Fhe United States' population y u = 225e° °
07
', and Mexico's population y M = 62e° °
24
'. When they are
the same, 225e° °
07 '= 62e° °
24
', 3.6290= e
0017 ', 0.017? = In 3.629== 1.2890, / = 75.82. Hence, the populations would be the same in the year 2055.
A bacterial culture, growing exponentially, increases from 100 to 400 grams in 10 hours. How much was present
after 3 hours?
y = 100e
K> . Hence, 400=100e
10K , 4 = e
lOK , 2 = e
5K , 5K = In 2 = 0.6931, AT = 0.13862. After
3 hours, y = 100e
3K = 100e°
41586 = 100(1.5156) = 151.56.
The population of Russia in 1980 was 255 million and growing exponentially with a growth constant of 0.012.
The population of the United States in 1980 was 225 million and growing exponentially with a growth constant of
0.007. When will the population of Russia be twice as large as that of the United States?
The population of Russia is y R = 225e°'
012 ' and that of the United States is y v = 225e° °
07
'. When the
population of Russia is twice that of the United States, 255e° °
12 '= 450e° °°
7
', e°
005 ' = 1.7647, 0.005f =
In 1.7647 = 0.56798. /«113.596 years; i.e., in the year 2093.
A bacterial culture, growing exponentially, increases from 200 to 500 grams in the period from 6 a.m. to 9 a.m.
How many grams will be present at noon?
217
26.20
26.21
26.22
26.23
26.24
26.25
26.26
26.27
26.28
26.29
If y is the number of flies y = 20e
005 '. When the enclosure is full, y = 640. Hence, 640 = 20e
005 ',
32 =e
005 ', 0.05f = In 32 = In (2
s ) = 5 In 2 = 3.4655. Hence, r = 69.31. Thus, it will take a little more than 69
days to fill the enclosure.
The number of bacteria in a certain culture is growing exponentially. If 100 bacteria are present initially and 400
are present after 1 hour, how many bacteria are present after 31 hours?
The growth equation is y = 100e
Kt . The given information tells us that 400 = 100e
K
, 4 = e
K . At
t = 3.5, y = !OOe
3iK = 100(e*)
3 5 = 100(4)
7
'
2 = 100(2)
7 = 100(128) = 12,800.
A certain radioactive substance has a half-life of 3 years. If 10 grams are present initially, how much of the
substance remains after 9 years?
Since 9 years is three half-lifes, ( \ )
3
10 = 1.25 grams will remain.
If y represents the amount by which the temperature of a body exceeds that of the surrounding air, then the rate at
which y decreases is proportional to y (Newton's law of cooling). Ify was initially 8 degrees and was 7 degrees
after 1 minute, what will it be after 2 minutes?
Since D,y = Ky, y = y 0 e
K> = 8e
Kl . The given facts tell us that 7 = 8e
K , e
K =\. When f = 2, y =
8e
2K = 8(e
K )
2 = 8(I)
2 = 6.125 degrees.
When a condenser is discharging electricity, the rate at which the voltage V decreases is proportional to V. If the
decay constant is K = -0.025, per second, how long does it take before V has decreased to one-quarter of its
initial value?
V=V 0 e~°025 '. When V is one-quarter of its initial value, \V 0 = V 0 e'
0 '
025 ', \=e~°025 ', -0.025f =
In J = -In 4 =-2 In 2 = -1.3862. Hence, f = 55.448.
The mass y of a growing substance is 7(5)' grams after t minutes. Find the initial quantity and the growth constant
K.
y = 7(5)' = l(e
1 "
5 )' = 7e
(ln 5)
'. When t = 0, y = 7 grams is the initial quantity. The growth constant K
is In 5.
If the population of Latin America has a doubling time of 27 years, by what percent does it grow per year?
The population y = y 0 e
Kl . By the given information, 2y 0 = y 0 e
27K , 2=e
27A: , e
K = V2« 1.0234. By
the solution to Problem 26.6, the percentage increase per year r = 100(e* -1) = 100(1.0234 - 1) = 2.34.
If in 1980 the population of the United States was 225 million and increasing exponentially with a growth constant
of 0.007, and the population of Mexico was 62 million and increasing exponentially with a growth constant of
0.024, when will the two populations be equal if they continue to grow at the same rate?
Fhe United States' population y u = 225e° °
07
', and Mexico's population y M = 62e° °
24
'. When they are
the same, 225e° °
07 '= 62e° °
24
', 3.6290= e
0017 ', 0.017? = In 3.629== 1.2890, / = 75.82. Hence, the populations would be the same in the year 2055.
A bacterial culture, growing exponentially, increases from 100 to 400 grams in 10 hours. How much was present
after 3 hours?
y = 100e
K> . Hence, 400=100e
10K , 4 = e
lOK , 2 = e
5K , 5K = In 2 = 0.6931, AT = 0.13862. After
3 hours, y = 100e
3K = 100e°
41586 = 100(1.5156) = 151.56.
The population of Russia in 1980 was 255 million and growing exponentially with a growth constant of 0.012.
The population of the United States in 1980 was 225 million and growing exponentially with a growth constant of
0.007. When will the population of Russia be twice as large as that of the United States?
The population of Russia is y R = 225e°'
012 ' and that of the United States is y v = 225e° °
07
'. When the
population of Russia is twice that of the United States, 255e° °
12 '= 450e° °°
7
', e°
005 ' = 1.7647, 0.005f =
In 1.7647 = 0.56798. /«113.596 years; i.e., in the year 2093.
A bacterial culture, growing exponentially, increases from 200 to 500 grams in the period from 6 a.m. to 9 a.m.
How many grams will be present at noon?