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) = 0.69(1.03)x
Part A: Is the price of product A increasing or decreasing and by what percentage per year?
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)
10,100 10,201 10,303.01 10,406.04
Which product recorded a greater percentage change in price over the previous year?
Part A:
To determine if the price of product A is increasing or decreasing, we need to analyze the function f(x) = 0.69(1.03)x.
Notice that the term (1.03)x is raising the decimal value 1.03 to the power of x. Since 1.03 is greater than 1, this means that the value inside the parentheses will increase as x increases.
Therefore, the whole function f(x) will also increase as x increases, which indicates that the price of product A is increasing.
To find the percentage by which the price is increasing per year, we need to calculate the percentage change in price for each year. We can start by comparing the price after x years to the price after (x - 1) years.
Let's calculate the percentage change for the first year:
f(1) = 0.69(1.03)^1 = 0.69(1.03) ≈ 0.7107 (rounded to four decimal places)
The percentage change can be calculated as: ((f(1) - f(0)) / f(0)) * 100.
f(0) represents the price after 0 years, which we can substitute into the formula:
f(0) = 0.69(1.03)^0 = 0.69(1) = 0.69
Now we can calculate the percentage change:
((0.7107 - 0.69) / 0.69) * 100 ≈ 2.99% (rounded to two decimal places)
Therefore, the price of product A is increasing by approximately 2.99% 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 each product.
Let's look at the table for product B:
t (number of years)
1 2 3 4
f(t) (price in dollars)
10,100 10,201 10,303.01 10,406.04
From the table, we can determine the percentage change for each year by using the formula ((f(t) - f(t-1)) / f(t-1)) * 100.
For the first year,
((10,201 - 10,100) / 10,100) * 100 ≈ 1%
For the second year,
((10,303.01 - 10,201) / 10,201) * 100 ≈ 1%
For the third year,
((10,406.04 - 10,303.01) / 10,303.01) * 100 ≈ 1%
Comparing the percentage changes for product B over the previous year, we see that the percentage change is approximately 1% for each year.
Since we found earlier that the percentage change for product A is approximately 2.99%, we can conclude that product A recorded a greater percentage change in price over the previous year compared to product B.