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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.

Sagot :

GinnyAnswer
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.
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