Applications Using Exponential Functions
This section will discuss applications involving exponential functions. While there
are many applications that use exponential functions, we will examine two –
compound interest and depreciation. These applications involve formulas. Thus,
understanding each formula and how to use them is important.
Compound Interest
Simple interest means that interest accrues based on the principal of an investment
or loan. The simple interest is calculated as a percent of the principal. When interest
is compounded, the interest is added to the principal, resulting in more interest
because the principal on which the interest is calculated increases each year.
The formula for interest earned when compounded annually is given by
A = P(1 + r)t
where A is the resulting amount in the investment, P is the principal, r is the interest
rate, and t is the time in years.
Example 1: Suppose $2,000 is invested in a bank account, and it earns 5% interest
annually. How much will be in the account after 3 years?
P = 2000
r = 0.05
t=3
A = 2000(1 + 0.05)3
A = 2000(1.05)3
A = 2000 × 1.157625
A = 2315.25
Substitute the given values into the formula
$2,315.25 will be in the account after 3 years
Modified from Intermediate Algebra, by Andrew Gloag, Anne Gloag, and Mara Landers, CK-12
Foundation, CC-BY 2013. Licensed under a Creative Commons Attribution 3.0 Unported License
(http://creativecommons.org/licenses/by/3.0)
Example 2: Suppose $2,000 is invested in a bank account at 5% interest
compounded annually. How much will be in the account after 20 years?
P = 2000
r = 0.05
t = 20
A = 2000(1 + 0.05)20
Substitute the given values into the formula
20
A = 2000(1.05)
A = 2000 × 2.653297705
A = 5306.60
Rounded to the nearest cent
$5,306.60 will be in the account after 20 years
Both examples above had the same principal amount and the same interest rate.
Graphing the function A = 2000(1.05)t, we can see the values for any number of
years.
Modified from Intermediate Algebra, by Andrew Gloag, Anne Gloag, and Mara Landers, CK-12
Foundation, CC-BY 2013. Licensed under a Creative Commons Attribution 3.0 Unported License
(http://creativecommons.org/licenses/by/3.0)
Interest is often compounded more frequently than annually, such as quarterly,
monthly, etc. In this case, the interest rate is divided by the number of compound
periods, and the time is multiplied by the number of compound periods. In general,
if interest is compounded n times per year, the formula is:
r nt
A = P(1 + )
n
Example 3: Suppose $5,000 is invested in an account at 6% interest compounded
quarterly. How much will be in the account after 10 years?
P = 5000
r = 0.06
n = quarterly = 4
t = 10
A = 5000(1 +
0.06 4×10
4
)
A = 5000(1 + 0.015)40
A = 5000(1.015)40
A = 5000 × 1.814018409
A = 9070.09
Substitute the values in the formula
Rounded to the nearest cent
$9,070.09 will be in the account after 10 years
Depreciation
While compound interest is an example of an increasing exponential function,
depreciation is an example of a decreasing exponential function. Over time, most
items are not worth as much as when they were new. This decrease in value is called
depreciation.
Modified from Intermediate Algebra, by Andrew Gloag, Anne Gloag, and Mara Landers, CK-12
Foundation, CC-BY 2013. Licensed under a Creative Commons Attribution 3.0 Unported License
(http://creativecommons.org/licenses/by/3.0)
The formula for the depreciated value of a vehicle is given by
V = P(1 – r)t
where V is the depreciated value, P is the original value, r is the rate of depreciation,
and t is time in years.
Example 4: A car purchased for $25,000 depreciates at a constant rate of 15%. What
will be the value of the car in 5 years?
V = 25000(1 – 0.15)5
V = 25000(0.85)5
V = 25000 × 0.443705313
V = 11092.63
Substitute the values into the formula
Rounded to the nearest cent
The value of the car in 5 years will be $11,092.63
Modified from Intermediate Algebra, by Andrew Gloag, Anne Gloag, and Mara Landers, CK-12
Foundation, CC-BY 2013. Licensed under a Creative Commons Attribution 3.0 Unported License
(http://creativecommons.org/licenses/by/3.0)