Exponential Growth & Decay Equations & Examples

An unknown radioactive element decays into non-radioactive substances. In 720 days the radioactivity of a sample decreases by 33 percent.
(a) What is the half-life of the element? half-life:_____
(6) How long will it take for a sample of 100 mg to decay to 47 mg? time needed:____

Describe what the values of C and k represent in the exponential growth and decay model ${\displaystyle y=C{e}^{kt}}$.

Rewrite the function to determine whether it represents exponential growth or exponential decay.
${\displaystyle y=2{\left(1.06\right)}^{9t}}$

The half-life of a radioactive kind of xenon is 9 hours. If you start with 32 grams of it, how much will be left after 18 hours?

Solve problems involving antiditerentiation
Solve situational problems involving exponential growth and decay
1. The rate of decay of radium is said to be proportional to the amount of radium present. If the half-life of radium is 1690 years and there are 200 grams on hand now, how much radium will be present in 845 years?

Using the concept of exponential growth and exponential decay, solve the given problem. Show complete and systematic solutions.
1. An unknown radioactive element decreases 12% of its amount every 5 days. What is exponential function? that describes the amount left after t days? If there are 300g of the substance at present, how much is left after 30 days? (round off the value up to two decimal places)

Tell whether the function represents exponential growth or exponential decay. Then graph the function.
${\displaystyle y={\left(1.8\right)}^x}$

Solve the following problems involving exponential growth and decay.
The half-life of carbon-14 1s approximately 6000 years. How much of 800 g of this substance will remain after 30,000 years?

Exponential Growth and Decay
Exponential growth and decay problems follow the model given by the equation ${\displaystyle A\left(t\right)=P{e}^{rt}}$
-The model is a function of time t
-A(t) is the amount we have ater time t
-PIs the initial amount, because for t=0, notice how ${\displaystyle A\left(0\right)=P{e}^{0\times t}=P{e}^0=P}$
-Tis the growth or decay rate. It is positive for growth and negative for decay
Growth and decay problems can deal with money (interest compounded continuously), bacteria growth, radioactive decay. population growth etc.
So A(t) can represent any of these depending on the problem.
Practice
The growth of a certain bactenia population can be modeled by the function
${\displaystyle A\left(t\right)=900e^{0.0534}}$
where A(t) is the number of bacteria and t represents the time in minutes.
How long will t take for the number of bacteria to double? (your answer must be accurate to at least 3 decimal places.)

Does a group with exponential growth always have a hyperbolic subgroup which has exponential growth?

How do you Find exponential decay rate?

Explain when a function in the form
${\displaystyle y=a\times b^x}$
models exponential growth and when it models exponential decay.

A certain radioactive substance has a half-life of 12 days. This means that every 12 days, half of the original amount of the substance decays. If there are 128 milligrams of the radioactive substance today, how many milligrams will be left after 48 days?

Exponential Growth and Decay
Exponential growth and decay problems follow the model given by the equation ${\displaystyle A\left(t\right)=P{e}^{rt}}$
-The model is a function of time t
-A(t) is the amount we have ater time t
-PIs the initial amount, because for t=0, notice how ${\displaystyle A\left(0\right)=P{e}^{0\times t}=P{e}^0=P}$
-Tis the growth or decay rate. It is positive for growth and negative for decay
Growth and decay problems can deal with money (interest compounded continuously), bacteria growth, radioactive decay. population growth etc.
So A(t) can represent any of these depending on the problem.
Practice
The growth of a certain bactenia population can be modeled by the function
${\displaystyle A\left(t\right)=900e^{0.0534}}$
where A(t) is the number of bacteria and t represents the time in minutes.
What is the initial number of bacteria? (round to the nearest whole number of bacteria.)

Exponential Growth /Decay Model
Given exponential growth or decay the amoun tP after time tis given by the following formula:
${\displaystyle P=P_0{e}^{kt}}$
Here ${\displaystyle P_0}$ is the initial amount and k is the exponential growth/decay rate.
Now you are ready to complete the following:
a) As stated previously the statement - The rate of change of variable y is proportional to the value of y-describes a differential equation. Write this differential equation.

Communicate Precisely How are exponential growth functions similar to exponential decay functions? How are they different?

In the exponential growth or decay function, explain the circumstances that cause k to be positive or negative.

Prove that if ${\displaystyle y=y^{0ekt},}$ where ${\displaystyle y_0}$ and k are constants, then ${\displaystyle \frac{dy}{dt}=ky}$
(This says that for exponential growth and decay, the rate of change of the population is proportional to the size of the population, and the constant of proportionality is the growth or decay constant.)