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.
### 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.
