Question

1. the data gives the estimated value in dollars of a model of a classic car over several years. 15300 16100 17300 18400 19600 20700 22000 find an approximate exponential model for the cars value by averaging the successive ratios of the value. you can solve by using a scientific calculator or use the exponential regression tool.

Explanation:

Step1: Calculate first ratio

$\frac{16100}{15300}\approx1.0523$

Step2: Calculate second ratio

$\frac{17300}{16100}\approx1.0745$

Step3: Calculate third ratio

$\frac{18400}{17300}\approx1.0636$

Step4: Calculate fourth ratio

$\frac{19600}{18400}\approx1.0652$

Step5: Calculate fifth ratio

$\frac{20700}{19600}\approx1.0561$

Step6: Calculate sixth ratio

$\frac{22000}{20700}\approx1.0628$

Step7: Calculate average ratio

$b=\frac{1.0523 + 1.0745+1.0636+1.0652+1.0561+1.0628}{6}\approx1.0624$

Step8: Determine initial value

$a = 15300$ (when $x = 0$)

Step9: Write exponential model

$y=15300(1.0624)^{x}$

Answer:

Let the exponential model be $y = ab^{x}$, where $y$ is the value of the car, $x$ is the number of years, $a$ is the initial - value, and $b$ is the growth factor.

First, find the successive ratios:

• Ratio 1: $\frac{16100}{15300}\approx1.0523$
• Ratio 2: $\frac{17300}{16100}\approx1.0745$
• Ratio 3: $\frac{18400}{17300}\approx1.0636$
• Ratio 4: $\frac{19600}{18400}\approx1.0652$
• Ratio 5: $\frac{20700}{19600}\approx1.0561$
• Ratio 6: $\frac{22000}{20700}\approx1.0628$

The average of these ratios is:

$$
LATEXBLOCK0
$$

Let $x = 0$ correspond to the first - data point, so $a = 15300$.

The exponential model is $y=15300(1.0624)^{x}$