Word Lesson: Exponential Growth

In order to solve problems involving exponential growth, it is necessary to

know how to solve an exponential equation
know the properties of logarithms
analyze and understand the problem
write and solve an equation for the problem

Exponential growth is generally applied to word problems such as compound interest problems and population growth problems. To grow exponentially means that the topic being studied is increasing in proportion to what was previously there. For example, money deposited in the bank earns interest that is added to the money previously in the bank.

Suppose you want to know how long will it take $1200 to double if it is invested at

compounded continuously.

First, we will need to use the exponential growth formula for compounding interest:

In the formula, A represents the amount of money that will be in the account when $1200 is doubled. P represents principal - the amount of money currently being invested. The letter r stands for rate of interest, and t stands time in years. In this formula e represents the irrational number 2.71828…..

Now, we need to substitute known values for the variables in the formula. The problem asks how long it will take $1200 to double. Therefore, A is 2400 (the value of 1200 doubled) in this problem. P is the money to be invested, so P is 1200. The rate, r, is which is or 0.105 as a decimal. Time t is what we are trying to find. So we have the following:

Finally we must solve the equation for time t. To do so, first divide both sides by 1200 to simplify the equation.

Now, we take the natural log of each side of the equation. For a reminder on taking the log of both sides as well as the properties of logs, please examine the material in this companion lesson.

Using the following property of logs, , we have:

Since the is equal to 1, we will substitute 1 for to give us the following:

Use a calculator to find the value for the ln 2 and then divide each side by 0.105 to obtain the final answer:

Therefore, we have determined that if $1200 is invested at compounded continuously, it will take 6.6 years for the money to double.

Examples

	Growth of bacteria in food products causes a need to “time-date” some products (like milk) so that shoppers will buy the product and consume it before the number of bacteria grows too large and the product goes bad. Suppose that the formula represents the growth of bacteria in a food product. The variable t represents time in days and represents the number of bacteria in millions. If the product cannot be eaten after the bacteria count reaches 4,000,000, how long will it take?

	What is your answer?

	Victor wants to buy a new car that costs $90,000. He has saved $20,000. Determine how many years it will take his $20,000 to grow to $90,000 at 6% interest compounded continuously.

	What is your answer?

Examples

	In a given year, the minimum wage was only $1.60 per hour. Use the exponential growth formula to predict when that minimum wage in the United States will reach 8.50 per hour if the rate of growth in the minimum wage is 3.9%. In the formula 1.6 represents a current minimum wage; A represents the amount of minimum wage you wish to obtain, and t represents time in years. 4.3 years 42.8 years 136.2 years -42.8 years

	What is your answer?

	Scientific research has shown that the risk of having a car accident increases exponentially as the concentration of alcohol in the blood increases. A formula that models the risk of an accident is the following: . In the formula, R represents the % of risk. [R will be given as a percent and should be used as a percent rather than a decimal in working the problem.] Find the blood alcohol concentration ( ) that corresponds to a 25% risk of a car accident. -0.25 11.15 -0.11 0.11

	What is your answer?