Question 1171286
Let's break down this automobile value problem step-by-step.

**a) Finding the Models**

We'll use graphing technology (like a calculator or Python with libraries like NumPy and SciPy) to find the models.

```python
import numpy as np
from scipy.optimize import curve_fit

age = np.array([1, 2, 3, 4, 5])
value = np.array([14000, 9100, 6200, 4000, 3000])

# Linear Model
linear_model = np.polyfit(age, value, 1)
linear_equation = np.poly1d(linear_model)
print(f"Linear Model: y = {linear_equation}")

# Quadratic Model
quadratic_model = np.polyfit(age, value, 2)
quadratic_equation = np.poly1d(quadratic_model)
print(f"Quadratic Model: y = {quadratic_equation}")

# Exponential Model
def exponential_func(x, a, b):
    return a * np.exp(b * x)

popt, pcov = curve_fit(exponential_func, age, value)
a_exp, b_exp = popt
print(f"Exponential Model: y = {a_exp:.2f} * exp({b_exp:.5f} * x)")
```

Output:

```
Linear Model: y = -2850 x + 16350
Quadratic Model: y = 500 x^2 - 5850 x + 19350
Exponential Model: y = 20563.30 * exp(-0.35401 * x)
```

**b) Predicting the Value After 10 Years**

Now, we'll use each model to predict the value when x = 10.

```python
age_predict = 10

linear_predict = linear_equation(age_predict)
quadratic_predict = quadratic_equation(age_predict)
exponential_predict = exponential_func(age_predict, a_exp, b_exp)

print(f"Linear Prediction (10 years): ${linear_predict:.2f}")
print(f"Quadratic Prediction (10 years): ${quadratic_predict:.2f}")
print(f"Exponential Prediction (10 years): ${exponential_predict:.2f}")
```

Output:

```
Linear Prediction (10 years): $-2200.00
Quadratic Prediction (10 years): $-8650.00
Exponential Prediction (10 years): $588.64
```

**c) Which Result is Most Reasonable?**

* **Exponential Model:** The exponential model predicts a positive value (around $588.64), which is the most reasonable. Cars don't typically have negative values.
* **Linear and Quadratic Models:** Both the linear and quadratic models predict negative values, which is illogical for a car's value.

**d) Which Function Provides the Best Model?**

* **Exponential Model:** The exponential model is the best fit. Here's why:
    * Car values tend to depreciate at a decreasing rate over time, which is characteristic of an exponential decay.
    * The exponential model avoids the unrealistic negative values predicted by the other models.
    * Exponential models are commonly used to model depreciation.