​Use the information about car values on page 564 to explain how you can use, assignment help

Use the information about car values on

page 564

to explain how you can use exponential decay to determine the current value of a car. Include a description of how to find the percent decrease in the value of the car each year and a description of how to find the value of a car for any given year when the rate of depreciation is known.

9-6
Exponential Growth and
Decay
Main Ideas
• Use logarithms to
solve problems
involving exponential
decay.
• Use logarithms to
solve problems
involving exponential
growth
New Vocabulary
rate of decay
rate of growth
GET READY for the Lesson
Certain assets, like homes, can
appreciate or increase in value over
time. Others, like cars, depreciate
or decrease in value with time.
Suppose you buy a car for $22,000
and the value of the car decreases
by 16% each year. The table shows
the value of the car each year
for up to 5 years after it
was purchased.
Years after
Purchase
0
1
2
3
4
5
Value of
Car ($)
22,000.00
18.480.00
15,523.20
13,039.49
10,953.17
9200.66
Exponential Decay The depreciation of the value of a car is an example
of exponential decay. When a quantity decreases by a fixed percent each
year, or other period of time, the amount y of that quantity after t years is
given by y -a(1 – 1), where a is the initial amount and r is the percent
of decrease expressed as a decimal. The percent of decrease r is also
referred to as the rate of decay.
EXAMPLE Exponential Decay of the Form y = (1-)
CAFFEINE A cup of coffee contains 130 milligrams of caffeine. If
caffeine is eliminated from the body at a rate of 11% per hour, how
long will it take for half of this caffeine to be eliminated?
Explore The problem gives the amount of caffeine consumed and the
rate at which the caffeine is eliminated. It asks you to find the
time it will take for half of the caffeine to be eliminated.
Plan Use the formula y = (1 – 1). Lett be the number of hours
Study Tip
since drinking the coffee. The amount remaining y is half of
130 or 65.
Rate of Change
Solve y = (1 – Exponential decay formula
Remember to rewrite
65 = 130(1 – 0.11)’ Replace y with 65, a with 130, and r with 11% or 0.11.
the rate of change as a
decimal before using it
0.5 = (0.89 Divide each side by 180
in the formula
log 0.5 = log (0.89) Property of Equality for Logarithms
log 0.5 = t log (0.89) Power Property for Logarithms
log 0.5
log 0.89
Divide each side by log 0.89
59480
Use a calculator
It will take approximately 6 hours.
544 Chapter 9 Exponential and Logarithmic Relations
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