Applications of Exponential Functions (base e)
Exponential functions arise when the rate of change of a variable depends on the current level of the variable. The exponential function then predicts what the level of the variable will be after a given interval of time.
Several typical examples of this phenomenon are often given:
- the rate at which a hot object cools depends on how hot it is in comparison with its surroundings. (This is called Newton's Law of Cooling.) We can predict the temperature of the object after a time interval $\Delta t$Δt.
- The rate at which a mass of radioactive material decays depends on the mass of the material present. Radioactive dating techniques depend on knowing how much radioactive material was originally present and how much is currently present. The rate of decay for the isotope in question is known and therefore, the time interval, the age of the material, can be calculated.
- The number of cells in a biological specimen has a particular doubling time, at least in the initial stages. Thus, the rate of increase depends on the number of cells already present and an exponential function can be constructed that predicts the number of cells that will be present after a given time.
The precise manner in which an expression for a rate of change is converted into an explicit formula for the level of some quantity comes from the study of differential equations. This is an important application of the branch of mathematics called calculus that was invented in the 17th century independently by Isaac Newton and Gottfried Leibniz.
Practice questions
Question 1
Under certain climatic conditions the proportion $P$P of the current blue-green algae population to the initial population satisfies the equation $P=e^{0.007t}$P=e0.007t, where $t$t is measured in days from when measurement began.
Solve for $t$t, the number of days it takes the initial number of algae to double to the nearest two decimal places.
Enter each line of work as an equation.
Question 2
The proportion $Q$Q of radium remaining after $t$t years is given by $Q=e^{-kt}$Q=e−kt, where $k$k is a constant.
After $1679$1679 years, only half the initial amount of radium remains.
Solve for $k$k.
Solve for $t$t, the number of years it takes for only $10%$10% of the initial amount of radium to remain to the nearest two decimal places.
Question 3
The remains of a human body can be dated by measuring the proportion of radiocarbon in tooth enamel.
The proportion of radiocarbon $A$A remaining $t$t years after a human passes away is given by $A=e^{-kt}$A=e−kt, where $k$k is a positive constant.
Solve for the value of $k$k if the amount of radiocarbon present is halved every $5594$5594 years.
Enter each line of work as an equation. Leave your answer in exact form.
For a particular corpse, the amount of radiocarbon present is only $25%$25% of the original amount at death. How many years ago, $t$t, did the person pass away?
Give your answer to two decimal places.