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)
Fuel A is .04 or 4% OF THE FIRST YEAR increase each year (simple interest)
for fuel B
3.22 / 3.04 = 1.06 or 6% increase per month (3.61/3.41 is also 6% )
This is not only higher interest percent but is COMPOUNDED
3.61/3.04 = 1.188 during 3 years or total of 6.2% per year if it were simple interest
Part A:
To determine whether the price of fuel A is increasing or decreasing and by what percentage per month, we can analyze the function f(x) = 2.96(1.04)^x.
The term (1.04)^x in the function represents the growth factor for each month. Since 1.04 is greater than 1, it indicates that each month the price is increasing.
To calculate the percentage increase per month, we can find the difference between f(x) for consecutive months and divide it by the previous month's price. Let's calculate the percentage change from month 1 to month 2:
f(1) = 2.96(1.04)^1 = 3.0784
f(2) = 2.96(1.04)^2 = 3.206144
Percentage change = ((f(2) - f(1)) / f(1)) * 100
= ((3.206144 - 3.0784) / 3.0784) * 100
= (0.127744 / 3.0784) * 100
≈ 4.15%
So, the price of fuel A is increasing by approximately 4.15% per month.
Part B:
To determine which type of fuel recorded a greater percentage change in price over the previous month, we need to compare the percentage changes in the prices of fuel A and fuel B.
Let's calculate the percentage change for fuel B over the previous month (from month 3 to month 4):
g(3) = 3.41
g(4) = 3.61
Percentage change for fuel B = ((g(4) - g(3)) / g(3)) * 100
= ((3.61 - 3.41) / 3.41) * 100
= (0.20 / 3.41) * 100
≈ 5.87%
Now, let's compare this with the percentage change for fuel A. We found earlier that the percentage change for fuel A is approximately 4.15% per month.
Therefore, the percentage change in price for fuel B over the previous month (5.87%) is greater than the percentage change for fuel A (4.15%).
Thus, 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, let's examine the function f(x) = 2.96(1.04)^x.
In this equation, the base, 1.04, is greater than 1. This indicates that each month, the price of fuel A will be multiplied by a number greater than 1.
Since the exponent x is positive (representing months), raising a number greater than 1 to a positive exponent will result in a number greater than 1.
Therefore, the price of fuel A is increasing over time.
To determine the percentage increase per month, we can calculate the percent change in price between two consecutive months.
Percent increase = [(New price - Old price) / Old price] * 100
Let's consider the change from month 1 to month 2:
New price = f(2) = 2.96(1.04)^2 = 3.075584
Old price = f(1) = 2.96(1.04)^1 = 3.0784
Plugging these values into the percent increase formula:
Percent increase = [(3.075584 - 3.0784) / 3.0784] * 100 ≈ -0.0914%
Therefore, the price of fuel A is increasing by approximately -0.0914% per month.
Part B:
To determine which type of fuel recorded a greater percentage change in price over the previous month, let's analyze the given data for fuel B.
Month 1 to Month 2:
Percent increase = [(3.22 - 3.04) / 3.04] * 100 ≈ 5.92%
Month 2 to Month 3:
Percent increase = [(3.41 - 3.22) / 3.22] * 100 ≈ 5.90%
Month 3 to Month 4:
Percent increase = [(3.61 - 3.41) / 3.41] * 100 ≈ 5.87%
Comparing these values, we can see that the percentage change in price for fuel B remains relatively consistent, with the highest being approximately 5.92%.
Since the percentage change for fuel A was approximately -0.0914%, it is significantly lower than any of the percentage changes for fuel B.
Therefore, fuel B recorded a greater percentage change in price over the previous month compared to fuel A.