Select the correct answer.

Adrianna is using exponential functions to model the value, in whole dollars, of two investments. She represents the value of investment A with a description of its key features and the value of investment B with a table. In both cases, [tex]\( x \)[/tex] is the number of years she has held the investment.

Investment A:
- Initial value: 2,000
- Average rate of change over the interval [tex]\([0,3]\)[/tex]: -95.1

Investment B:
[tex]\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
0 & 1,800 \\
\hline
1 & 1,710 \\
\hline
2 & 1,624 \\
\hline
3 & 1,543 \\
\hline
\end{array}
\][/tex]

Which statement is true about the two investments during the first three years?
A. Both investments are decreasing in value, and investment B is decreasing in value faster.
B. Investment A is increasing in value, but investment B is decreasing in value.
C. Both investments are decreasing in value, and investment A is decreasing in value faster.
D. Both investments are decreasing in value at the same average rate.



Answer :

To determine which statement is true about the two investments during the first three years, let's analyze the given information and find the average rates of change for both investments.

### Investment A:

- The initial value of investment A is 2,000 dollars.
- The average rate of change for investment A over the interval [tex]\([0, 3]\)[/tex] is -95.1 dollars per year.

An average rate of change of -95.1 indicates that investment A is decreasing in value because the rate of change is negative.

### Investment B:

- The table provides the values of investment B at different years:
[tex]\[ \begin{array}{|c|c|} \hline x & g(x) \\ \hline 0 & 1800 \\ \hline 1 & 1710 \\ \hline 2 & 1624 \\ \hline 3 & 1543 \\ \hline \end{array} \][/tex]

To find the average rate of change for investment B over the interval [tex]\([0, 3]\)[/tex]:
[tex]\[ \text{Average Rate of Change for B} = \frac{g(3) - g(0)}{3 - 0} \][/tex]
Substituting the values from the table:
[tex]\[ \text{Average Rate of Change for B} = \frac{1543 - 1800}{3} = \frac{-257}{3} \approx -85.67 \text{ dollars per year} \][/tex]

An average rate of change of approximately -85.67 indicates that investment B is also decreasing in value because the rate of change is negative.

### Comparison:

Since both investments have a negative average rate of change, both investments are decreasing in value over the first three years.

- For investment A, the average rate of decrease is -95.1 dollars per year.
- For investment B, the average rate of decrease is approximately -85.67 dollars per year.

Given that the magnitude of the average rate of decrease for investment A ([tex]\(|-95.1|\)[/tex]) is greater than that of investment B ([tex]\(|-85.67|\)[/tex]), investment A is decreasing in value faster than investment B.

### Conclusion:

Based on this analysis, the correct statement is:

C. Both investments are decreasing in value, and investment A is decreasing in value faster.