To solve this problem, we need to determine which exponential function best fits the given data on the loan balance over time.
First, let's identify the possible forms of the exponential functions:
[tex]\[ \text{A. } f(x) = 20{,}500 \cdot 1.11^x \][/tex]
[tex]\[ \text{B. } f(x) = 20{,}500 \cdot 1.05^x \][/tex]
[tex]\[ \text{C. } f(x) = 20{,}000 \cdot 1.005^x \][/tex]
[tex]\[ \text{D. } f(x) = 20{,}000 \cdot 0.095^x \][/tex]
Next, we'll analyze the data provided:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Months} & \text{Loan Balance} \\
\hline
6 & \$20{,}608 \\
\hline
12 & \$21{,}234 \\
\hline
18 & \$21{,}879 \\
\hline
24 & \$22{,}543 \\
\hline
30 & \$23{,}228 \\
\hline
36 & \$23{,}934 \\
\hline
\end{array}
\][/tex]
We assume the relationship is exponential, which generally has the form:
[tex]\[ f(x) = a \cdot b^x \][/tex]
Given the loan started at \$20{,}000, the initial amount [tex]\(a\)[/tex] can be directly identified as:
[tex]\[ a = 20{,}000 \][/tex]
To determine the base [tex]\(b\)[/tex] of the exponential function, we perform a logarithmic transformation of the loan balance data and fit a linear regression to this transformed data.
From the processed data, the base [tex]\( b \)[/tex] and initial slope [tex]\( m \)[/tex] are given by:
[tex]\[ b \approx 1.005 \][/tex]
[tex]\[ \text{slope} \approx 0.004987 \][/tex]
This gives us the exponential function:
[tex]\[ f(x) = 20{,}000 \cdot (1.005)^x \][/tex]
Now let's compare the candidate exponential functions provided:
- Option A: [tex]\( f(x) = 20{,}500 \cdot 1.11^x \)[/tex]
- A significant deviation from the identified function with [tex]\(a = 20{,}000\)[/tex] and [tex]\(b \approx 1.005\)[/tex].
- Option B: [tex]\( f(x) = 20{,}500 \cdot 1.05^x \)[/tex]
- Also deviates significantly due to both a different initial amount and base.
- Option C: [tex]\( f(x) = 20{,}000 \cdot 1.005^x \)[/tex]
- This matches the derived function closely.
- Option D: [tex]\( f(x) = 20{,}000 \cdot 0.095^x \)[/tex]
- This is an entirely different relation and doesn't match exponent or initial amount.
Among the options provided, the best fit is:
[tex]\[ \boxed{f(x)=20,000 \cdot 1.005^x} \][/tex]
Hence, the correct option is:
[tex]\[ \boxed{C} \][/tex]
