2.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) = 72(1.25)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)
65 84.5 109.85 142.81
Which product recorded a greater percentage change in price over the previous year? Justify your answer.
Part A: To determine if the price of product A is increasing or decreasing, we need to look at the growth rate represented by the function f(x) = 72(1.25)^x.
In this function, the base of the exponent, 1.25, is greater than 1. This means that the function is showing exponential growth.
Therefore, the price of product A is increasing over time.
To calculate the percentage increase per year, we need to find the growth factor. The growth factor is the value by which the price increases each year. In this case, it is given by the constant multiplier, 1.25.
To find the percentage increase per year, we subtract 1 from the growth factor and multiply by 100:
Percentage increase per year = (1.25 - 1) * 100%
Percentage increase per year = 0.25 * 100%
Percentage increase per year = 25%
Therefore, the price of product A is increasing by 25% 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 change in price for each product.
For product B, we have the following prices over the years:
t (number of years)
1 2 3 4
f(t) (price in dollars)
65 84.5 109.85 142.81
To find the percentage change in price over the previous year for each year, we can use the formula:
Percentage change = ((f(t) - f(t-1)) / f(t-1)) * 100%
Using this formula, we can calculate the percentage change in price for each year and compare them.
For year 2:
Percentage change = ((84.5 - 65) / 65) * 100%
Percentage change ≈ (19.5 / 65) * 100%
Percentage change ≈ 30%
For year 3:
Percentage change = ((109.85 - 84.5) / 84.5) * 100%
Percentage change ≈ (25.35 / 84.5) * 100%
Percentage change ≈ 30%
For year 4:
Percentage change = ((142.81 - 109.85) / 109.85) * 100%
Percentage change ≈ (32.96 / 109.85) * 100%
Percentage change ≈ 30%
From the calculations, we can see that product B recorded a constant 30% increase in price over the previous year for each year.
Since we already determined that the price of product A increases by 25% per year, and product B consistently increases by 30% over the previous year for each year, we can conclude that product B has a greater percentage change in price over the previous year.
Part A: To determine if the price of product A is increasing or decreasing and by what percentage per year, we can analyze the given function f(x) = 72(1.25)^x.
In the function f(x) = 72(1.25)^x:
- The base, 1.25, is greater than 1, indicating exponential growth.
- As x increases, the exponent increases, resulting in an exponential increase in the value of (1.25)^x.
- The coefficient 72 is a scaling factor that determines the initial price.
Therefore, based on the function, the price of product A is increasing over time.
To calculate the percentage increase per year, we can consider the difference in prices over one year and express it as a percentage. The percentage increase formula is:
Percentage Increase = ((New Value - Old Value) / Old Value) * 100.
Let's calculate the percentage increase in price for product A after 1 year:
Old Value = f(0) = 72(1.25)^0 = 72
New Value = f(1) = 72(1.25)^1 = 90
Percentage Increase = ((90 - 72) / 72) * 100 ≈ 25%
Therefore, the price of product A is increasing by approximately 25% per year.
Part B: To determine which product recorded a greater percentage change in price over the previous year, we can analyze the given table for product B.
Using the price values in the table, we can calculate the percentage change in price over each year.
For Year 1: Percentage Increase = ((84.5 - 65) / 65) * 100 ≈ 29.23%
For Year 2: Percentage Increase = ((109.85 - 84.5) / 84.5) * 100 ≈ 30.00%
For Year 3: Percentage Increase = ((142.81 - 109.85) / 109.85) * 100 ≈ 29.99%
From the calculations, we can see that product B recorded a greater percentage change in price over the previous year compared to product A, which had an approximately constant 25% increase per year.
Therefore, product B experienced a greater percentage change in price over the previous year than product A.
You know that this function is DECREASING because 0.63, the number inside of the parenthesis, is LESS THAN THE NUMBER 1. I [think I] know the way to determine by what percentage it is decreasing. [I think] My teacher taught me that I have to subtract the number, in this case 0.63, from 1. So, 1 - 0.63 = 0.37. So it is decreasing by 0.37.
B: Year 1-2 they are all changing by the same percentage: 43% less. And you can prove it by dividing year 2 to year 1 and dividing year 3 to year 2.