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A pair of women's running shoes cost [tex]$130 in 2004 and $[/tex]155 in 2009 due to inflation.

a. Develop an exponential model to describe the rate of inflation over this time period.
b. If inflation continues at the same rate, use your model in part (a) to estimate the price of the shoes in 2014.

a. The rate of inflation can be modeled by the following exponential equation, where the model solves for the inflated price [tex]\( y \)[/tex] after time [tex]\( x \)[/tex].
[tex]\[ y = 130 \left(\square^x\right) \][/tex]
(Round to four decimal places as needed.)

Sagot :

a. Developing an Exponential Model:

To model the rate of inflation, we can use the concept of exponential growth. The general form of the exponential growth equation is:

[tex]\[ y = P (1 + r)^t \][/tex]

where:
- [tex]\( y \)[/tex] is the final amount ([tex]$155 in 2009), - \( P \) is the initial amount ($[/tex]130 in 2004),
- [tex]\( r \)[/tex] is the rate of inflation,
- [tex]\( t \)[/tex] is the number of years.

From the question, we know that the shoes cost [tex]$130 in 2004 and $[/tex]155 in 2009. Therefore, we have:

[tex]\[ 155 = 130 (1 + r)^{2009 - 2004} \][/tex]
[tex]\[ 155 = 130 (1 + r)^5 \][/tex]

First, solve for [tex]\( (1 + r) \)[/tex]:

[tex]\[ \frac{155}{130} = (1 + r)^5 \][/tex]
[tex]\[ 1.1923 = (1 + r)^5 \][/tex]

To solve for [tex]\( r \)[/tex], we take the fifth root of both sides:

[tex]\[ 1 + r = \left(1.1923\right)^{1/5} \approx 1.0358 \][/tex]

Thus, the rate of inflation [tex]\( r \)[/tex] is:

[tex]\[ r = 1.0358 - 1 \approx 0.0358 \][/tex]

So, the exponential model describing the inflation rate is:

[tex]\[ y = 130 (1.0358)^x \][/tex]

b. Estimating the Price in 2014:

Now, we need to estimate the price of the shoes in 2014 using the exponential model developed in part (a).

Using the model [tex]\( y = 130 (1.0358)^x \)[/tex], we need to find the price when [tex]\( x \)[/tex] is the number of years from 2004 to 2014:

[tex]\[ x = 2014 - 2004 = 10 \][/tex]

Substitute [tex]\( x = 10 \)[/tex] into the model:

[tex]\[ y = 130 (1.0358)^{10} \][/tex]

Using the calculation, we get:

[tex]\[ y \approx 130 \times 1.348 \][/tex]
[tex]\[ y \approx 175.32 \][/tex]

Therefore, the estimated price of the shoes in 2014, given the continuous rate of inflation, is approximately:

[tex]\[ \$ 184.81 \][/tex]