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 whether the price of product A is increasing or decreasing, we can analyze the given function: f(x) = 12500(0.82)^x.
In this function, the base (0.82) is less than 1. When a base of an exponential function is less than 1, it indicates decay or decrease.
So, the price of product A is decreasing over time.
To determine the percentage decrease per year, we need to compare the initial price to the price after one year, f(1):
f(1) = 12500(0.82)^1 = 10250
To find the percentage decrease, we can calculate the difference between the initial price and the price after one year, and divide it by the initial price:
Percentage decrease = (12500 - 10250) / 12500 * 100
= 2250 / 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 find the percentage change for both products.
For product B, we are given the prices after each year: f(1) = 5600, f(2) = 3136, f(3) = 1756.16, f(4) = 983.45.
To find the percentage change in price from year n to (n-1), we can use the formula:
Percentage change = ((f(n) - f(n-1)) / f(n-1)) * 100.
Calculating the percentage change for each year, we get:
Percentage change from year 2 to year 1 = ((3136 - 5600) / 5600) * 100 = -43.71%
Percentage change from year 3 to year 2 = ((1756.16 - 3136) / 3136) * 100 = -43.97%
Percentage change from year 4 to year 3 = ((983.45 - 1756.16) / 1756.16) * 100 = -43.96%
From the above calculations, we can see that the percentage change in price over the previous year is the same (-43.97%) for each year.
Therefore, both products, A and B, recorded the same percentage change in price over the previous year.