4.6 Exponential Growth and Decay
Solve the given problem related to population growth.
A city had a population of 22,300 in 2000 and a population of 25,200 in 2005.
(a) Find the exponential growth function for the city. Use t = 0 to represent 2000. (Round k to five decimal places.)
(b) Use the growth function to predict the population of the city in 2015. Round to the nearest hundred.
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Solve the given problem related to population growth.
The population of Charlotte, North Carolina, is growing exponentially.
The population of Charlotte was 395,934 in 1990 and 610,949 in 2005. Find the exponential growth function that models the population of Charlotte. Use t = 0 to represent 1990. (Round k to ten decimal places.)
Use it to predict the population of Charlotte in 2013. Round to the nearest thousand.
Sodium­24 is a radioactive isotope of sodium that is used to study circulatory dysfunction. Assuming that 6 micrograms of sodium­24 is injected into a person, the amount A in micrograms remaining in that person after t hours is given by the equation A = 6e­0.046t.
(a) Graph this equation.
(b) What amount of sodium­24 remains after 2 hours? (Round your answer to two decimal places.)
(c) What is the half­life of sodium­24? (Round your answer to two decimal places.)
(d) In how many hours will the amount of sodium­24 be 1 microgram? (Round your answer to two decimal places.)
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Use the half­life information from Table 4.11, page 392, to work the exercise.
Geologists have determined that Crater Lake in Oregon was formed by a volcanic eruption. Chemical analysis of a wood chip assumed to be from a tree that died during the eruption has shown that it contains approximately 30% of its original carbon­14. Estimate how long ago the volcanic eruption occurred. (Round your answer to the nearest year.)
Use the half­life information from Table 4.11, page 392, to work the exercise.
Estimate the age of a bone if it now contains 65% of its original amount of carbon­14. Round to the nearest 100 years.
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If $4,000 is invested at an annual interest rate of 3% and compounded annually, find the balance after the following number of years. (Round your answers to the nearest cent.)
(a) 4 years
(b) 7 years
Find the balance if $20,000 is invested at an annual rate of 10% for 5 years, compounded continuously. (Round your answer to the nearest cent.)
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Determine the following constants for the given logistic growth model.
(a) the carrying capacity
(b) the growth rate constant
(c) the initial population P0 (Round your answer to the nearest whole number.)
Solve the given problem related to population growth.
The population of squirrels in a nature preserve satisfies a logistic model in which P0 = 2800 in 2007. The carrying capacity of the preserve is estimated at 7200 squirrels, and P(2) = 3200.
(a) Determine the logistic model for this population, where t is the number of years after 2007. (Round logistic model parameters to five decimal places.)
(b) Use the logistic model from (a) to predict the year in which the squirrel population will first exceed 5900.
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Newton's Law of Cooling states that if an object at temperature T0 is placed into an environment at constant temperature A, then the temperature of the object, T(t) (in degrees Fahrenheit), after t minutes is given by T(t) = A + (T0 − A)e­kt,
where k is a constant that depends on the object.
(a) Determine the constant k (to the nearest thousandth) for a canned soda drink that takes 5 minutes to cool from 71°F to 57°F after being placed in a refrigerator that maintains a constant temperature of 34°F.
(b) What will be the temperature (to the nearest degree) of the soda drink after 30 minutes?
(c) When (to the nearest minute) will the temperature of the soda drink be 44°F?
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