Sure, let's break down this problem step by step for each part:
### Part A
We need to determine whether the price of fuel A is increasing or decreasing and by what percentage per month.
The price of fuel A after [tex]\( x \)[/tex] months is given by the function:
[tex]\[ f(x) = 2.27(0.88)^x \][/tex]
1. Identify whether the price is increasing or decreasing:
The term [tex]\( (0.88)^x \)[/tex] indicates an exponential function. Since the base of the exponent is [tex]\( 0.88 \)[/tex] which is less than 1, the function represents exponential decay.
Therefore, the price of fuel A is decreasing.
2. Calculate the percentage decrease per month:
For an exponential decay situation of the form [tex]\( f(x) = a(b)^x \)[/tex], where [tex]\( 0 < b < 1 \)[/tex], the decay rate per period can be calculated as:
[tex]\[ \text{Decay rate per month} = (1 - b) \times 100 \% \][/tex]
Here, [tex]\( b = 0.88 \)[/tex]:
[tex]\[ \text{Decay rate per month} = (1 - 0.88) \times 100 \% = 0.12 \times 100 \% = 12 \% \][/tex]
So, the price of fuel A is decreasing by 12% per month.
### Part B
We have the prices for fuel B over 4 months given in a table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
\text{m (number of months)} & 1 & 2 & 3 & 4 \\
\hline
\text{g(m) (price in dollars)} & 3.44 & 3.30 & 3.17 & 3.04 \\
\hline
\end{array}
\][/tex]
We need to determine which type of fuel recorded a greater percentage change in price over the previous month. To do this, we will first calculate the percentage change in price for fuel B for each month:
1. Calculate the percentage change from month 1 to month 2:
[tex]\[
\text{Percentage change} = \left( \frac{3.30 - 3.44}{3.44} \right) \times 100 \% = -4.07 \%
\][/tex]
2. Calculate the percentage change from month 2 to month 3:
[tex]\[
\text{Percentage change} = \left( \frac{3.17 - 3.30}{3.30} \right) \times 100 \% = -3.94 \%
\][/tex]
3. Calculate the percentage change from month 3 to month 4:
[tex]\[
\text{Percentage change} = \left( \frac{3.04 - 3.17}{3.17} \right) \times 100 \% = -4.10 \%
\][/tex]
Therefore, the percentage changes for each month for fuel B are approximately [tex]\(-4.07\% \)[/tex], [tex]\(-3.94\% \)[/tex], and [tex]\(-4.10\% \)[/tex].
4. Average percentage change for fuel B:
[tex]\[
\text{Average percentage change} = \frac{-4.07 - 3.94 - 4.10}{3} \approx -4.03 \%
\][/tex]
5. Determine the highest absolute percentage change for fuel B:
The highest absolute percentage change for fuel B is [tex]\(|-4.10 \%|\)[/tex] which is [tex]\(4.10 \%\)[/tex].
### Conclusion
- For fuel A, the price is decreasing by 12% per month.
- For fuel B, the percentage changes over the months were approximately [tex]\(-4.07\% \)[/tex], [tex]\(-3.94\% \)[/tex], [tex]\(-4.10\%\)[/tex]. The greatest monthly percentage change for fuel B is [tex]\(|-4.10\%| = 4.10 \%\)[/tex].
Thus, comparing the changes:
- Fuel A recorded a greater percentage change of [tex]\(12\%\)[/tex] per month in comparison to fuel B's greatest change of [tex]\(4.10\%\)[/tex] per month.
