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Core Connections Integrated I, 2013

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Core Connections Integrated I, 2013 View details

2. Section 11.2

Continue to next subchapter

Exercise 62 Page 607

Practice makes perfect

a Since the stock's price decreases by a percentage, we can write the stock's value as an exponential function.

y = a b^{x}

In this function,

a

is the initial value and

b

is the multiplier. From the exercise we know the stock's value in

2005

and

2010 .

If we let

2005

be the starting year, we can interpret the information as the data points

(0, 60)

and

(5, 45) .

If we substitute these points in the function, we get two equations.

{60 = a b^{0} 45 = a b^{5}

By solving this system of equations we can determine the multiplier.

{60 = a b^{0} 45 = a b^{5} (I) (II)

$(I):$ $a^{0} = 1$

{60 = a \cdot 1 45 = a b^{5}

$(I):$ $a \cdot 1 = a$

{60 = a 45 = a b^{5}

$(I):$ Rearrange equation

{a = 60 45 = a b^{5}

$(II):$ $a = 60$

{a = 60 45 = (60) b^{5}

▼

(II):

Solve for

b

$(II):$ $LHS / 60 = RHS / 60$

{a = 60 \frac{4 5}{6 0} = b^{5}

$(II):$ $\frac{a}{b} = \frac{a / 1 5}{b / 1 5}$

{a = 60 \frac{3}{4} = b^{5}

$(II):$ Rearrange equation

{a = 60 b^{5} = \frac{3}{4}

$(II):$ $5 LHS = 5 RHS$

{a = 60 b = 5 \frac{3}{4}

$(II):$ Use a calculator

{a = 60 b = 0.94408 \dots

$(II):$ Round to $3$ decimal place(s)

{a = 60 b \approx 0.944

The multiplier is about

0.944,

which corresponds to a percentage decrease of

5.6 % .

b From Part B we calculated the initial value and multiplier to be

a = 60

and

b \approx 0.944,

respectively. Let's substitute these values into the general form of an exponential function.

y = 60 (0.944)^{x}

c The year

2020

is

15

years after the starting year of

2005 .

Therefore, by substituting

x = 15

into the function we wrote in Part B we can predict the value in the year

2020 .

y = 60 (0.944)^{x}

$x = 15$

y = 60 (0.944)^{15}

Use a calculator

y = 25.27732 \dots

Round to $2$ decimal place(s)

y \approx 25.28

The value in

2020

should be about

$ 25.28 .

Subchapter links

11.2.1 Solving Work and Mixture Problems p.603-604

11.2.2 Solving Equations and Systems Graphically p.607-608

11.2.3 Using a Best-Fit Line to Make a Prediction p.610-614

11.2.4 Treasure Hunt p.616-617

11.2.5 Using Coordinate Geometry and Constructions to Explore Shapes p.620-621

11.2.6 Modeling with Exponential Functions and Linear Inequalities p.625-628