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Continuous Growth and Decay

Author: Sophia

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what's covered

In this lesson, you will investigate continuous growth. Specifically, this lesson will cover:

1. Continuous Growth and Decay
- 1a. The Exponential Model for Continuous Growth and Decay
- 1b. The Relationship Between Continuous Growth/Decay and Percent Increase/Decrease
2. Continuous Compounding

1. Continuous Growth and Decay

1a. The Exponential Model for Continuous Growth and Decay

In calculus, you will be able to study growth and decay rates at specific points in time, rather than between two points in time. When a quantity is growing continuously, we say that the rate at which the quantity is changing is proportional to the quantity present. The constant of proportionality is called the continuous growth or decay rate.

Since calculus is required to determine a formula for the quantity present at time t, the continuous growth/decay formula is given next.

formula to know

Exponential Growth/Decay

The equation A open parentheses t close parentheses equals a times e to the power of r t end exponent comma

for all real numbers t and where

is a positive constant, is the continuous growth/decay formula, where:

the initial value
r = the continuous growth rate (percent written in decimal form)
t = the elapsed time (no specific units needed as long as they are kept consistent)

The value of r is used to determine whether this is a growth or decay model:

If then this formula represents continuous growth.
If then this formula represents continuous decay.

Here is an example to get us started.

EXAMPLE

Consider the equation A open parentheses t close parentheses equals 500 e to the power of 0.06 t end exponent.

The initial value is 500.

Since r equals 0.06

is positive, the continuous growth rate is 0.06, or 6%.

try it

Consider the equation A open parentheses t close parentheses equals 20 e to the power of short dash 0.124 t end exponent.

What is the initial value?

Does this equation describe continuous growth or decay, and why?

This equation describes decay since r equals short dash 0.124 comma

which is negative.

What is the continuous growth/decay rate?

The continuous decay rate is 12.4%.

term to know

Continuous Growth/Decay Rate

The rate at which a quantity is changing at one specific point in time.

1b. The Relationship Between Continuous Growth/Decay and Percent Increase/Decrease

Using properties of exponents, we can actually link the continuous growth or decay rate with its corresponding percent increase or decrease.

EXAMPLE

Consider the equation A open parentheses t close parentheses equals 500 e to the power of 0.06 t end exponent comma

where t is measured in months.

By using properties of exponents, we can write A open parentheses t close parentheses equals 500 open parentheses e to the power of 0.06 end exponent close parentheses to the power of t.

A open parentheses t close parentheses equals 500 open parentheses e to the power of 0.06 end exponent close parentheses to the power of t.

This means that A open parentheses t close parentheses

is an exponential model of the form A open parentheses t close parentheses equals a times b to the power of t

, where

and

b equals e to the power of 0.06 end exponent.

Since

e to the power of 0.06 end exponent almost equal to 1.061837 comma

the growth rate is approximately 6.1837% each month, while the original version of the model shows a continuous growth rate of 6%.

This means that a continuous growth rate of 6% per month corresponds to an increase of approximately 6.1837% each month.

Let’s look at a decay problem before pulling some ideas together.

EXAMPLE

Consider the equation A open parentheses t close parentheses equals 20 e to the power of short dash 0.124 t end exponent comma

where t is the time in years.

By using properties of exponents again, we have A open parentheses t close parentheses equals 20 open parentheses e to the power of short dash 0.124 end exponent close parentheses to the power of t.

A open parentheses t close parentheses equals 20 open parentheses e to the power of short dash 0.124 end exponent close parentheses to the power of t.

This means that A open parentheses t close parentheses

is an exponential model of the form A open parentheses t close parentheses equals a times b to the power of t

, where

and

b equals e to the power of short dash 0.124 end exponent.

Since

e to the power of short dash 0.124 end exponent almost equal to 0.88338 comma

we find the yearly percent decrease as follows:

	Subtract the base from 1.
11.662%	Convert to percent form.

This means that a continuous decay rate of 12.4% per year is equivalent to a decrease of 11.662% per year.

In both examples, you may notice that similar calculations were performed to find the percent increase in each time period.

Given a continuous growth or decay formula, perform the following steps to find the corresponding percent increase or decrease for each unit of time.

step by step

Calculate the value of
Now, compare the value of to 1.
1. If this indicates growth. Calculate and convert to a percent. This is the corresponding percent increase for each unit of time.
2. If this indicates decay. Calculate and convert to a percent. This is the corresponding percent decrease for each unit of time.

watch

This video demonstrates an example of finding the corresponding percent increase or decrease for A open parentheses t close parentheses equals 1 fifth e to the power of 0.047 t end exponent.

A open parentheses t close parentheses equals 1 fifth e to the power of 0.047 t end exponent.

try it

Consider the exponential decay model A open parentheses t close parentheses equals 30 e to the power of short dash 0.08 t end exponent comma

where t is measured in years.

Calculate the corresponding percent decrease each year. Round your final answer to two decimal places.

In this case, the value of e to the power of r

e to the power of short dash 0.08 end exponent comma

which is approximately 0.9231163464…, which is less than 1.

Therefore, this model represents decay. The decay rate is 1 minus 0.923116... almost equal to 0.076884.

1 minus 0.923116... almost equal to 0.076884.

To convert to a percent, multiply by 100 (or simply move the decimal point two spaces to the right) to get 7.6884…%. Rounded to the nearest hundredth of a percent, the decay rate is 7.69%.

This shows that the model A open parentheses t close parentheses equals a times e to the power of r t end exponent can be written as f open parentheses t close parentheses equals a times b to the power of t comma and vice versa. While we are able to find the percent increase/decrease given the value of r, we still need to learn more mathematics in order to find the value of r that corresponds to a given percent increase/decrease.

Here is an example of how we can apply the continuous growth/decay equation.

EXAMPLE

Radon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of radon-222 decay to in three days?

First, start with the equation, which is A open parentheses t close parentheses equals a times e to the power of r t end exponent colon

A open parentheses t close parentheses equals a times e to the power of r t end exponent colon

Since the starting value is 100 mg,
Since the continuous rate of decay is 17.3%,

Then, the equation that models this situation is A open parentheses t close parentheses equals 100 e to the power of short dash 0.173 t end exponent.

Now, we want to know the amount present after three days, which means t equals 3.

Thus, to the nearest hundredth, A open parentheses 3 close parentheses equals 100 e to the power of short dash 0.173 open parentheses 3 close parentheses end exponent almost equal to 59.51 space mg.

A open parentheses 3 close parentheses equals 100 e to the power of short dash 0.173 open parentheses 3 close parentheses end exponent almost equal to 59.51 space mg.

try it

Strontium-90 decays at a continuous rate of 2.41% per year.

How much will 500 mg of strontium-90 decay to in six years? Round to the nearest hundredth.

Since the continuous decay rate is 2.41%, and the initial amount is 500 mg, the amount remaining is modeled by the function A open parentheses t close parentheses equals 500 e to the power of short dash 0.0241 t end exponent comma

A open parentheses t close parentheses equals 500 e to the power of short dash 0.0241 t end exponent comma

where t is the number of years.

To find the amount present after 6 years, find A open parentheses 6 close parentheses colon

A open parentheses 6 close parentheses equals 500 e to the power of short dash 0.0241 open parentheses 6 close parentheses end exponent almost equal to 432.68 space mg

2. Continuous Compounding

Consider an investment of $100,000 that is invested for one year in an account that pays an annual interest rate of 2%. Let’s compare the amount available at the end of one year for various compounding intervals:

Type of Compounding	n = # Compounding Periods Per Year	Accumulated Amount (Nearest Cent)
Quarterly	4	$100000 open parentheses 1 plus fraction numerator 0.02 over denominator 4 end fraction close parentheses to the power of 4 equals $ 102 comma 015.05$
Monthly	12	$100000 open parentheses 1 plus fraction numerator 0.02 over denominator 12 end fraction close parentheses to the power of 12 equals $ 102 comma 018.44$
Daily	365	$100000 open parentheses 1 plus fraction numerator 0.02 over denominator 365 end fraction close parentheses to the power of 365 equals $ 102 comma 020.08$
Hourly	8,760	$100000 open parentheses 1 plus fraction numerator 0.02 over denominator 8760 end fraction close parentheses to the power of 8760 equals $ 102 comma 020.13$
Every Minute	525,600	$100000 open parentheses 1 plus fraction numerator 0.02 over denominator 525600 end fraction close parentheses to the power of 525600 equals $ 102 comma 020.13$

Notice that this process could be carried out every second, millisecond, etc., until we are seemingly adding interest continuously. If you examine the values in the last column of the table, you see that the amount available after one year seems to be leveling off as the number of compounding periods per year increases. This is, in fact, the case.

This “upper limit” is found by compounding continuously. In business applications, the continuous growth/decay formula can be adapted to give the continuous compounding formula.

formula to know

Compounding Continuously

A open parentheses t close parentheses equals P e to the power of r t end exponent comma

where:

P = the principal (initial investment)
r = the annual interest rate
t = the time in years that the money was invested

Let’s use this formula to check our results in the last example.

EXAMPLE

Given a principal of $100,000 invested at an interest rate of 2% compounded continuously, let's find the amount available after one year.

First, identify the quantities:

Then, the continuous compounding function is A open parentheses t close parentheses equals 100000 e to the power of 0.02 t end exponent.

Now, substitute t equals 1

into the formula, then use your calculator to approximate to the nearest cent.

A open parentheses 1 close parentheses equals 100000 e to the power of 0.02 open parentheses 1 close parentheses end exponent almost equal to $ 102 comma 020.13.

A open parentheses 1 close parentheses equals 100000 e to the power of 0.02 open parentheses 1 close parentheses end exponent almost equal to $ 102 comma 020.13.

To the nearest cent, this result is equal to the results obtained by compounding each minute and each hour.

try it

Suppose that $5,000 is invested into an account that pays interest at an annual rate of 3% compounded continuously.

How much is in the account after five years? Round to the nearest cent.

Start with the continuous compound interest formula, which is A equals P e to the power of r t end exponent comma

with

and

We have:

A equals 5000 e to the power of 0.03 open parentheses 5 close parentheses end exponent
space space space equals 5000 e to the power of 0.15 end exponent
space space space almost equal to 5809.17

The amount in the account at the end of five years is $5,809.17.

did you know

While not used in practice, continuous compounding is used by financial professionals simply because the formula is much easier to calculate, and it gives the maximum accumulated amount for a given principal, interest rate, and investment period.

summary

In this lesson, you learned that when the rate of change in a quantity is proportional to the quantity itself, the constant of proportionality is called the continuous growth or decay rate (depending on whether it is positive or negative). You learned that the exponential model for continuous growth and decay model is another way of representing an exponential function, one that reflects different information. You also learned about the relationship between continuous growth/decay and percent increase/decrease, which is that the continuous growth/decay rate corresponds to a percent increase/decrease over the same interval of time. Lastly, you learned that in business applications, the continuous growth/decay formula can be adapted to compound interest, resulting in the continuous compounding formula.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.

Terms to Know

Continuous Growth/Decay Rate: The rate at which a quantity is changing at one specific point in time.

Formulas to Know

Compounding Continuously: $A open parentheses t close parentheses equals P e to the power of r t end exponent comma$ where:
• P = the principal (initial investment)
• r = the annual interest rate
• t = the time in years that the money was invested
Exponential Growth/Decay: The equation $A open parentheses t close parentheses equals a times e to the power of r t end exponent comma$ for all real numbers t and where $a$ is a positive constant, is the continuous growth/decay formula, where:
• $a equals$ the initial value
• r = the continuous growth rate (percent written in decimal form)
• t = the elapsed time (no specific units needed as long as they are kept consistent)

The value of r is used to determine whether this is a growth or decay model:
• If $r greater than 0 comma$ then this formula represents continuous growth.
• If $r less than 0 comma$ then this formula represents continuous decay.

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Continuous Growth and Decay

Table of Contents

1. Continuous Growth and Decay

1a. The Exponential Model for Continuous Growth and Decay

1b. The Relationship Between Continuous Growth/Decay and Percent Increase/Decrease

2. Continuous Compounding