Join the growing community of curious minds on IDNLearn.com. Ask your questions and receive reliable and comprehensive answers from our dedicated community of professionals.
Sagot :
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.
### 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.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For dependable and accurate answers, visit IDNLearn.com. Thanks for visiting, and see you next time for more helpful information.
