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Core Connections Algebra 2, 2013

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Core Connections Algebra 2, 2013 View details

2. Section 6.2

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Exercise 138 Page 299

Practice makes perfect

a This is an example of exponential decay. This means that the car's value can be described by an exponential function with a multiplier

b

that is between

0

and

1 .

y = a b^{x}

Since the car decreases annually by

20 %,

we have a multiplier of

b = 0.8 .

y = a (0.8)^{x}

b From Part A, we have already written half the function. We are only missing the initial value. In Part B, we have been given this value as

a = 23500 .

y = 23500 (0.8)^{t}

c By substituting

t = 4

into the function from Part B, we can determine the car's worth in four years.

y = 23500 (0.8)^{t}

$t = 4$

y = 23500 (0.8)^{4}

Use a calculator

y = 9625.6

Round to $2$ significant digit(s)

y \approx 9600

The car is worth about

$ 9600

in four years.

d We find when the car has a trade-in value of

$ 6000

by substituting

6000

for

y

in the equation from Part B and solving for

t .

y = 23500 (0.8)^{t}

$y = 6000$

6000 = 23500 (0.8)^{t}

▼

Solve for

t

Rearrange equation

23500 (0.8)^{t} = 6000

$LHS / 23500 = RHS / 23500$

0 . 8^{t} = \frac{6 0 0 0}{2 3 5 0 0}

$lo g (LHS) = lo g (RHS)$

lo g 0 . 8^{t} = lo g \frac{6 0 0 0}{2 3 5 0 0}

$lo g (a^{m}) = m \cdot lo g (a)$

t lo g 0.8 = lo g \frac{6 0 0 0}{2 3 5 0 0}

$LHS / lo g 0.8 = RHS / lo g 0.8$

t = \frac{lo g \frac{6 0 0 0}{2 3 5 0 0}}{lo g 0 . 8}

Use a calculator

t = 6.11821 \dots

Round to $1$ decimal place(s)

t \approx 6.1

Thus, approximately

6.1

years from now the car will be worth

$ 6000 .

e When

t = 0,

the car's value is

$ 23500 .

We want to know what the car was worth when it was new, which was

2.7

years ago. We can find that by substituting

- 2.7

for

t

in the equation.

y = 23500 (0.8)^{t}

$t = - 2.7$

y = 23500 (0.8)^{- 2.7}

Use a calculator

y = 42926.44242 \dots

Round to $2$ significant digit(s)

y \approx 43000

The car was worth about

$ 43000

when it was new.

Subchapter links

6.2.1 Using Logarithms to Solve Exponential Equations p.284-285

6.2.2 Investigating the Properties of Logarithms p.290-291

6.2.3 Writing Equations of Exponential Functions p.294-295

6.2.4 An Application of Logarithms p.299-301