7.4 Exponential growth and decay
Under ideal conditions, the population grows in accordance with the law , where denotes the population initially present in the culture, k is some constant rate determined by population under consideration, and t is the elapsed time.
General example with the given relative growth rate of 2% per year then,
, and the express of the population
1. Suppose 10,000 bacteria are present initially in the culture and 60,000 present 2 hours later. How many bacteria will there be in the culture at the end of 4 hours?
2. Radioactive substances decay exponentially. For example, the amount of radium present at any time t obeys the law , where is the initial amount present and k is a suitable positive constant. The half-life of a radioactive substance is the time required for a given amount to be reduced by one-half. Now, it is known that the half-life of radium is approximately 1600 years. Suppose initially there are 200 milligrams of pure radium. Find the amount left after t years. What is the amount left after 100 years?
3. A skull from an archeological site has one-tenth the amount of C-14 that originally contained. Determine the approximate age of the skull.
4. Eastman’s training department determines that after completing the basic training program, a new, previously inexperienced employee will be able to assemble cameras per day, t months after the employee starts work on the assembly line.
a) How many cameras can a new employee assemble per day after basic training?
b) How many cameras can an employee with 1 month experience assemble per day? 2 months? 6 months?
c) How many cameras can the average experienced employee assemble per day?
5. The number of soldiers at Fort MacArthur who contracted influenza after t days during a flu epidemic is approximated by the exponential model
If 40 soldiers contracted the flu by day 7, find how many soldiers contracted the flu by day 15.
6. Three hundred students attended the dedication ceremony of a new building on a college campus. The president of the traditionally female college announced the new expansion program, which included plans to make the college coeducational. The number of students who learned of the new program t hr later is given by the function
If 600 students on campus had heard about the new program 2 hrs after the ceremony, how many students had heard about the policy after 4 hr?
1) The temperature of a mug of coffee after t minutes is given by
where T is measured in degrees Fahrenheit.
What is the initial temperature of the coffee?
Answer: 180o F
When (to the nearest hundredth of a minute) will the coffee be at 100o?
Answer: after 8.84 minutes
2. A quantity is described by the exponential growth function
, where t is measured in minutes.
What is the growth constant?
Answer: 0.05
What quantity is initially present?
Answer: 1200
What quantity is present after 10 minutes? Round your answer to the nearest unit
Answer: 1978
3) A radioactive element has a half-life of 400 years. What is the decay constant?
Answer: 0.001732868
4) During a flu epidemic, the number of children in a school district who contracted
influenza after t days is given by
How many children had contracted influenza after the first day?
Answer: 11
How many children had contracted influenza after five days?
Answer: 76
5) A city currently has a population of 150,000. A city planner estimates that the
population of the city will be 200,000 in 15 years. If this is true, what is the
annual rate of population growth?
Answer: 0.019 or 1.9%
6) The population growth of a certain rodent is approximately 3% per month. Find
the time it takes for the population to triple.
Answer: months
7) A quantity P(t) is described by the exponential growth function ,
where t is measured in hours.
(a) What is the initial quantity?
Answer: 1000
(b) What quantity is present after 5 hours?
Answer: about 1036
(c) How long does it take to reach 1200?
Answer: about 26 hour
8) A radioactive element has a half-life of 400 years. What is the decay constant?
Answer: 0.001732868
9) A radioactive substance has a half-life of 20 years. If 200 g of the substance are
present initially, find the amount in 18 years.
Answer: g
10) A nature preserve is being established. A population biologist has estimated that
the population of the deer in the preserve is currently 150 and will increase at an
annual growth rate of 5%.
Find the function that expresses the deer population as a function of time t (in years).
Answer:
(b) Estimate the deer population in 10 years.
Answer: 247
11) Wood deposits recovered from an archaeological site contain 35% of the carbon
14 they originally contained. How long ago did the tree from which the wood
was obtained die? The decay constant of carbon 14 is k = 0.00012.
Answer: 8749 years
12) The population of rabbits in a park is initially 120 and is expected to increase at
an annual rate of 5%. Find the expected population in 5 years.
Answer: 154 rabbits