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? 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) 10,100 10,201 10,303.01 10,406.04
Which product recorded a greater percentage change in price over the previous year? Justify your answer. (5 points
.69*1.03 = .7107
f(x) = .7107 x
so decreasing, depreciation wins
1-.7107 = .2893
so value decreasing by about 29% per year
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in 3 years
(10406.04 -10,100)/4 = 102 increase/year
102/10100 * 100 = 1.01 percent increase per year
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 function f(x) = 0.69(1.03)^x.
Step 1: Understanding the function:
In this function, x represents the number of years. The term 1.03 represents a multiplier indicating an annual growth rate of 3% (1 + 3% = 1.03). The term 0.69 is the initial price of product A.
Step 2: Analyzing the function:
Since the exponent x is positive, we can conclude that the value inside the parentheses (1.03) is greater than 1, indicating exponential growth.
Step 3: Determining if the price is increasing or decreasing:
Since the base of the exponent (1.03) is greater than 1, the value of the function f(x) will increase over time. Therefore, the price of product A is increasing.
Step 4: Calculating the percentage increase per year:
To calculate the percentage increase per year, we need to find the difference between the prices after x years and after (x-1) years, and then divide it by the price after (x-1) years. Using the formula:
Percentage increase = ((f(x) - f(x-1))/f(x-1)) * 100
Let's calculate the percentage increase for x = 1 to x = 2:
f(1) = 0.69(1.03)^1 = 0.69(1.03) = 0.71
f(2) = 0.69(1.03)^2 = 0.69(1.0609) = 0.73
Percentage increase from x = 1 to x = 2:
((f(2) - f(1))/f(1)) * 100 = ((0.73 - 0.71)/0.71) * 100 = (0.02/0.71) * 100 ≈ 2.82%
Therefore, the price of product A is increasing by approximately 2.82% per year.
Part B: To determine which product recorded a greater percentage change in price over the previous year, we compare the percentage increase for each product.
For product B, we can use the given table to determine the percentage change over the previous year:
For t = 1 to t = 2:
Percentage increase = ((f(2) - f(1))/f(1)) * 100 = ((10201 - 10100)/10100) * 100 = (101/10100) * 100 ≈ 1%
For t = 2 to t = 3:
Percentage increase = ((f(3) - f(2))/f(2)) * 100 = ((10303.01 - 10201)/10201) * 100 = (102.01/10201) * 100 ≈ 1%
For t = 3 to t = 4:
Percentage increase = ((f(4) - f(3))/f(3)) * 100 = ((10406.04 - 10303.01)/10303.01) * 100 = (103.03/10303.01) * 100 ≈ 1%
By comparing the percentage changes for product A (2.82%) and product B (1%), we can conclude that product A had a greater percentage change in price over the previous year.
Therefore, product A recorded a greater percentage change in price over the previous year.