The price of products may increase due to inflation and decrease due to depreciation. Derek is studying the change in the price of two products, A and B, over time.
The price f(x), in dollars, of product A after x years is represented by the function below:
f(x) = 12500(0.82)x
Part A: Is the price of product A increasing or decreasing and by what percentage per year? Justify your answer. (5 points)
Part B: The table below shows the price f(t), in dollars, of product B after t years:
t (number of years) 1 2 3 4
f(t) (price in dollars) 5600 3136 1756.16 983.45
Which product recorded a greater percentage change in price over the previous year? Justify your answer. (5 points)
Part A: To determine if the price of product A is increasing or decreasing and by what percentage per year, we need to analyze the given function f(x) = 12500(0.82)^x.
First, let's observe the base of the exponential function, which is 0.82. Since the base is less than 1, we can conclude that the function is a decreasing exponential function. This means that as 'x' (time) increases, the price of product A decreases.
To determine the percentage change per year, we need to find the difference between the initial price (f(0)) and the price after one year (f(1)), and then calculate the percentage change.
f(0) = 12500(0.82)^0 = 12500(1) = 12500
f(1) = 12500(0.82)^1 ≈ 10250
The percentage change is calculated by taking the difference (10250 - 12500) and dividing it by the initial price (12500), then multiplying by 100 to convert it into a percentage:
Percentage change = ((10250 - 12500) / 12500) * 100 ≈ -18%
Therefore, the price of product A is decreasing by approximately 18% per year.
Part B: To determine which product recorded a greater percentage change in price over the previous year, we need to compare the percentage changes for product A and product B.
For product B, we are given a table with the prices f(t) for different values of 't' (years). To find the percentage change over the previous year for each value of 't', we need to compare f(t) with f(t-1).
For example, to find the percentage change in price from year 2 to year 1:
Percentage change from year 2 to year 1 = ((3136 - 5600) / 5600) * 100 ≈ -44%
We can perform similar calculations for the other values of 't' to find the respective percentage changes.
Comparing the percentage changes calculated for product A and product B, we can determine which product recorded a greater percentage change over the previous year. The product with the higher percentage change value has experienced a greater change in price over the given time period.
Therefore, to answer part B of the question, we need the percentage change values for both products.