Exponential growth models are often used for real-world situations like interest earned on an investment, human or animal population, bacterial culture growth, etc.
The general exponential growth model is
y = C(1 + r)t,
where C is the initial amount or number, r is the growth rate (for example, a 2% growth rate means r = 0.02), and t is the time elapsed.
A population of 32,000 with a 5% annual growth rate would be modeled by the equation:
y = 32000(1.05)t
with t in years.
Sometimes, you may be given a doubling or tripling rate rather than a growth rate in percent. For example, if you are told that the number of cells in a bacterial culture doubles every hour, then the equation to model the situation would be:
y = C · 2t
with t in hours.
Exponential decay models are also used very commonly, especially for radioactive decay, drug concentration in the bloodstream, of depreciation of value.
Radioactive decay problems are often given in terms of half-life of a radioactive element. This is modeled by the equation:
where N0 is the initial amount of the element, N is the amount remaining after t years, and τ is the half-life.
If you start with a quantity of the unstable element Potassium-40, it takes 1.26 billion years for half of it to decay into Argon-40. So the half-life of Potassium-40 is 1.26 billion years.
Here, N0 = 2000 and τ = 1,260,000,000. So the model is:
For drug concentration problems, you may be given the fraction p of the original amount of the drug left in the bloodstream after a unit of time. In this case, the situation is modeled by the equation
y = Apt,
where y is the concentration remaining after time t, and A is the initial amount.
If a person takes A milligrams of a drug at time 0, then y = A(0.8)t gives the concentration left in the bloodstream after t hours. If the initial dose is 200 mg, what is the concentration of the drug in the bloodstream after 4 hours?
y = 200(0.8)4
You might want a calculator!
y = 200(0.4096)
y = 81.92
So there are about 82 milligrams of the drug left in the bloodstream after four hours.
If the value of some article (for example, a car), originally $C, depreciates x% per year, then the value after t years is given by the formula:
y = C(1 – x/100)t
The original value of a car is $28,000. If it depreciates by 15% each year, find its value in 4 years.
y = 28000(1 – 0.15)4
y = 28000(0.85)4
y = 28000(0.52200625)
y = 14616.175
So after four years, the car is worth about $14,616.
In the exponential function given below, identify the initial amount and the growth rate.
y = 250(1 + 0.2)t
y = 9.8(1.35)t
Write an exponential growth function to model the situation.
A population of 422,000 increases by 12% each year.
You start with $30,000 and earn 15% interest each year.
Percentage of increase = 15%
Number of years = 25
An initial population of 750 endangered turtles triples each year for 5 years. Find the growth factor for the population and the population after 5 years.
The population of Baconburg starts off at 20,000, and grows by 13% each year. Write an exponential growth model and find the population after 10 years.
A car bought for $13,000 depreciates at 12% per annum. What is its value after 7 years?
If a person takes A milligrams of a drug at time 0, then y = A(0.7) t gives the concentration left in the bloodstream after t hours. If the initial dose is 125 mg, what is the concentration of the drug in the bloodstream after 3 hours?
Does the equation y = 11(1.11) t model exponential growth or exponential decay?
Find the growth or decay factor and the percent change per time period.
Does the equation y = 27(3/2) t model exponential growth or exponential decay?
Does the equation y = 7(3/4) t model exponential growth or exponential decay?