Let's analyze the changes in fuel prices step-by-step:
### Part A:
Determine whether the price of fuel A is increasing or decreasing, and by what percentage per month.
Given the function for the price of fuel A:
[tex]\[ f(x) = 2.27(0.88)^x \][/tex]
The base of the exponential function is [tex]\(0.88\)[/tex], which is less than 1. This indicates that the price of fuel A is decreasing over time.
To determine the percentage decrease:
1. Calculate the monthly decay factor: Since [tex]\(0.88\)[/tex] is the factor by which the price decreases monthly, the percentage decrease is given by:
[tex]\[ \text{Percentage decrease} = (1 - 0.88) \times 100 \][/tex]
2. Compute the value:
[tex]\[ \text{Percentage decrease} = 0.12 \times 100 \][/tex]
[tex]\[ \text{Percentage decrease} = 12\% \][/tex]
Conclusion: The price of fuel A is decreasing by 12% per month.
### Part B:
Compare the percentage change in price for fuel B over the previous month.
Given the prices of fuel B over four months:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
m \, (\text{number of months}) & 1 & 2 & 3 & 4 \\
\hline
g(m) \, (\text{price in dollars}) & 3.44 & 3.30 & 3.17 & 3.04 \\
\hline
\end{array}
\][/tex]
We need to calculate the percentage change in price for each month:
1. From month 1 to month 2:
[tex]\[ \text{Percentage change} = \left( \frac{3.30 - 3.44}{3.44} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = \left( \frac{-0.14}{3.44} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = -4.07\% \][/tex]
2. From month 2 to month 3:
[tex]\[ \text{Percentage change} = \left( \frac{3.17 - 3.30}{3.30} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = \left( \frac{-0.13}{3.30} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = -3.94\% \][/tex]
3. From month 3 to month 4:
[tex]\[ \text{Percentage change} = \left( \frac{3.04 - 3.17}{3.17} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = \left( \frac{-0.13}{3.17} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage change} = -4.10\% \][/tex]
Average percentage change for fuel B:
[tex]\[ \text{Average percentage change} = \frac{-4.07 + -3.94 + -4.10}{3} \][/tex]
[tex]\[ \text{Average percentage change} = \frac{-12.11}{3} \][/tex]
[tex]\[ \text{Average percentage change} = -4.04\% \][/tex]
### Comparison:
Fuel A: Decreasing by 12% per month.
Fuel B: Decreasing on average by approximately 4.04% per month.
Conclusion: Fuel A recorded a greater percentage change in price over the previous month, with a 12% decrease compared to the approximately 4.04% decrease recorded for fuel B.
