To analyze the change in the function [tex]\( f(x) = 3 \cdot 1.55^x \)[/tex] over the interval from [tex]\( x = 6 \)[/tex] to [tex]\( x = 7 \)[/tex], we will follow these steps:
1. Calculate [tex]\( f(6) \)[/tex]:
[tex]\[
f(6) = 3 \cdot 1.55^6
\][/tex]
After performing the calculation, we find:
[tex]\[
f(6) \approx 41.60173504687501
\][/tex]
2. Calculate [tex]\( f(7) \)[/tex]:
[tex]\[
f(7) = 3 \cdot 1.55^7
\][/tex]
After performing this calculation, we obtain:
[tex]\[
f(7) \approx 64.48268932265627
\][/tex]
3. Determine the change in [tex]\( f(x) \)[/tex]:
[tex]\[
\Delta f(x) = f(7) - f(6)
\][/tex]
Substituting the values we've found:
[tex]\[
\Delta f(x) \approx 64.48268932265627 - 41.60173504687501 = 22.880954275781264
\][/tex]
4. Calculate the percentage increase:
[tex]\[
\text{Percentage increase} = \left( \frac{\Delta f(x)}{f(6)} \right) \times 100
\][/tex]
Substituting the values:
[tex]\[
\text{Percentage increase} = \left( \frac{22.880954275781264}{41.60173504687501} \right) \times 100 \approx 55.00000000000003\%
\][/tex]
Based on these calculations:
- [tex]\( f(x) \)[/tex] at [tex]\( x = 6 \)[/tex] is approximately 41.60173504687501.
- [tex]\( f(x) \)[/tex] at [tex]\( x = 7 \)[/tex] is approximately 64.48268932265627.
- The change in [tex]\( f(x) \)[/tex] is approximately 22.880954275781264.
- The percentage increase is approximately 55.0%.
Therefore, the correct statement is:
[tex]\[ f(x) \text{ increases by } 55\% \][/tex]
