QuestionModel the sales of electronic devices, $S(t)$ , since 1997 and find the annual growth rate from 1997 to 2009 when sales were 438 million.

Studdy Solution

STEP 1

Assumptions1. The sales of electronic devices doubled between1997 and2009. . In2009,438 million electronic devices were sold.
3. We are looking for an exponential function to model the sales.
4. The exponential function has the form $(t) =0 \times (1 + r)^t$ , where $0$ is the initial value, $r$ is the growth rate, and $t$ is the time in years.
5. The year1997 is considered as the starting point, i.e., $t=0$ .
6. The year2009 is $t=2009-1997=12$ years from the starting point.

STEP 2

First, we need to find the initial sales $0$ in1997. Since the sales doubled from1997 to2009, we can find the initial sales by dividing the sales in2009 by2.
$0 = \frac{Sales\, in\,2009}{2}$

STEP 3

Now, plug in the given value for the sales in2009 to calculate the initial sales.
$0 = \frac{438\, million}{2}$

STEP 4

Calculate the initial sales.
$0 = \frac{438\, million}{2} =219\, million$

STEP 5

Now that we have the initial sales, we can write the exponential function $(t)$ .
$(t) =0 \times (1 + r)^t$

STEP 6

Plug in the value for the initial sales $0$ .
$(t) =219\, million \times (1 + r)^t$ This is the exponential function that models the sales of electronic devices in the US since199.

STEP 7

To find the annual percentage growth rate, we need to find the value of $r$ . We can do this by setting $(t)$ equal to the sales in2009 when $t=12$ and solving for $r$ .
$438\, million =219\, million \times (1 + r)^{12}$

STEP 8

Divide both sides of the equation by $219\, million$ to isolate the term involving $r$ .
$(1 + r)^{12} = \frac{438\, million}{219\, million}$

STEP 9

Calculate the right side of the equation.
$( + r)^{12} = \frac{438\, million}{219\, million} =2$

STEP 10

Take the12th root of both sides of the equation to solve for $+ r$ .
$+ r = \sqrt[12]{2}$

STEP 11

Subtract from both sides of the equation to solve for $r$ .
$r = \sqrt[]{} -$

STEP 12

Calculate the value of $r$ .
$r = \sqrt[12]{2} - \approx0.05946$

STEP 13

To express the growth rate as a percentage, multiply $r$ by100.
$r\, (\%) =0.05946 \times100$

STEP 14

Calculate the annual percentage growth rate.
$r\, (\%) =0.05946 \times100 \approx.946\%$ The annual percentage growth rate between1997 and2009 was approximately.946%.

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