The price of fuel may increase due to demand and decrease due to overproduction. Marco is studying the change in the price of two types of fuel, A and B, over time.
The price f(x), in dollars, of fuel A after x months is represented by the function below:
f(x) = 2.96(1.04)x
Part A: Is the price of fuel A increasing or decreasing and by what percentage per month? Justify your answer. (5 points)
Part B: The table below shows the price g(m), in dollars, of fuel B after m months:
m (number of months) 1 2 3 4
g(m) (price in dollars) 3.04 3.22 3.41 3.61
Which type of fuel recorded a greater percentage change in price over the previous month? Justify your answer. (5 points)
Part A: To determine if the price of fuel A is increasing or decreasing and by what percentage per month, we can look at the function f(x) = 2.96(1.04)^x.
Since the base of the exponent is 1.04, which is greater than 1, we can conclude that the price of fuel A is increasing over time.
To calculate the percentage increase per month, we can find the difference between the prices of two consecutive months and divide it by the initial price, and then multiply by 100.
For example, if we consider the prices after one month and two months, we have:
Price after one month: f(1) = 2.96(1.04)^1 = 2.96 * 1.04 ≈ 3.07
Price after two months: f(2) = 2.96(1.04)^2 ≈ 3.19
The percentage increase from one month to two months is:
[(3.19 - 3.07) / 3.07] * 100 ≈ 3.90%
Therefore, the price of fuel A is increasing by approximately 3.90% per month.
Part B: Comparing the prices of fuel B over consecutive months, we can calculate the percentage change to determine the type of fuel that recorded a greater percentage change in price.
Percentage change from month 1 to month 2:
[(3.22 - 3.04) / 3.04] * 100 ≈ 5.92%
Percentage change from month 2 to month 3:
[(3.41 - 3.22) / 3.22] * 100 ≈ 5.90%
Percentage change from month 3 to month 4:
[(3.61 - 3.41) / 3.41] * 100 ≈ 5.87%
By comparing the percentage changes, we can see that fuel B recorded a greater percentage change in price over the previous month.
Part A:
To determine whether the price of fuel A is increasing or decreasing, we need to look at the growth factor in the function f(x) = 2.96(1.04)^x.
In this equation, the base, 1.04, represents the growth rate per month. A value greater than 1 indicates an increase, while a value less than 1 indicates a decrease.
Here, the base is 1.04, which is greater than 1. Therefore, the price of fuel A is increasing. The question also asks for the percentage change per month, which is given by the growth rate minus 1, multiplied by 100%.
So, the percentage change per month for fuel A is (1.04 - 1) * 100% = 4%. The price of fuel A is increasing by 4% per month.
Part B:
To determine which type of fuel recorded a greater percentage change in price over the previous month, we need to calculate the percentage change for each fuel.
For fuel B, we have the following prices over the previous months:
g(1) = $3.04
g(2) = $3.22
To calculate the percentage change, we use the formula:
Percentage change = ((New Value - Old Value) / Old Value) * 100%
For fuel B, the percentage change from month 1 to month 2 is calculated as:
((3.22 - 3.04) / 3.04) * 100% = (0.18 / 3.04) * 100% ≈ 5.92%
Now let's calculate the percentage change for fuel A over the previous month.
Using the function f(x) = 2.96(1.04)^x, we plug in the values for x = 1 and x = 2:
f(1) = 2.96(1.04)^1 ≈ 3.0816
f(2) = 2.96(1.04)^2 ≈ 3.2104
The percentage change from month 1 to month 2 for fuel A is:
((3.2104 - 3.0816) / 3.0816) * 100% ≈ 4.18%
Comparing the two percentages, we can see that fuel B had a greater percentage change in price over the previous month (5.92% > 4.18%). Therefore, fuel B recorded a greater percentage change in price.
1.04^x means it grows 4% per month
divide g(m+1)/g(m) to get the growth rate.