The price of products may increase due to inflation and decrease due to depreciation. Marco 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.
since 0.82 < 1 clearly the price is decreasing by .18 = 18% each year
3136/5600 = 0.56
so, what do you think?
Part A: The price of product A is decreasing over time. We can see this from the function, where the value of 0.82 is less than 1. As x increases, the value of 0.82 raised to the power of x becomes smaller, resulting in a decrease in price. To find the percentage change per year, we can calculate the difference between f(x) and f(x+1) and divide it by f(x):
Percentage change per year = ((f(x+1) - f(x)) / f(x)) * 100
Let's calculate it for two consecutive years:
f(1) = 12500(0.82)^1 ≈ 10250
f(2) = 12500(0.82)^2 ≈ 8411
Percentage change from year 1 to year 2 = ((8411 - 10250) / 10250) * 100 ≈ -20.15%
Therefore, the price of product A is decreasing by approximately 20.15% per year.
Part B: To find the percentage change in price over the previous year, we need to compare the prices recorded at each year.
Percentage change from year 1 to year 2 = ((3136 - 5600) / 5600) * 100 ≈ -44.57%
Percentage change from year 2 to year 3 = ((1756.16 - 3136) / 3136) * 100 ≈ -43.97%
Percentage change from year 3 to year 4 = ((983.45 - 1756.16) / 1756.16) * 100 ≈ -43.92%
Product B has a greater percentage change in price over the previous year than product A (which is approximately -20.15%). Therefore, product B recorded a greater percentage change in price over the previous year.
Note: While the values of percentage changes for product B are decreasing, they are still greater in magnitude than the percentage change in product A.
Part A:
To determine whether the price of product A is increasing or decreasing, we need to analyze the expression 0.82.
Since this expression is less than 1, we can conclude that the price of product A is decreasing over time.
To determine the percentage decrease per year, we need to examine the value of the exponent x.
Since there is no specific value mentioned for x, we can consider it to be a general representation of time in years.
Therefore, the percentage decrease per year can be calculated by taking the difference between 1 and 0.82 and multiplying it by 100 to convert it into a percentage.
Percentage decrease = (1 - 0.82) * 100 = 18%
So, the price of product A is decreasing by 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 price values of product A and product B for the given years.
Let's calculate the percentage change in price for each product over the previous year:
For product A:
The initial price of product A after 1 year is given as f(1) = 12500(0.82)^1 = 10250.
The price difference compared to the previous year is f(1) - f(0) = 10250 - 12500 = -2250.
The percentage change is (-2250/12500) * 100 = -18%.
For product B:
The initial price of product B after 1 year is given as f(1) = 5600.
The price difference compared to the previous year is f(1) - f(0) = 5600 - 0 = 5600.
The percentage change is (5600/0) * 100 = undefined (division by zero error).
From the calculations, we can see that the percentage change in price is greater for product A (-18%) compared to product B (undefined). Therefore, product A recorded a greater percentage change in price over the previous year.
Justification: The percentage change in price is calculated by comparing the difference in price between the current year and the previous year, and dividing it by the price in the previous year multiplied by 100. Since product B has a price of 0 in the previous year, dividing by 0 causes an undefined result. Thus, product A 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 and by what percentage per year, we need to analyze the function f(x) = 12500(0.82)^x.
The base of the exponential function, (0.82), is less than 1, which indicates that the product is experiencing depreciation. Typically, when the base is less than 1, the function represents exponential decay.
Therefore, the price of product A is decreasing over time. To determine the rate of decrease or the percentage change per year, we can calculate the difference between the initial price (f(0)) and the price at a later time (f(x)) and then divide it by the number of years (x).
Let's calculate the percentage decrease per year:
Initial price (f(0)) = 12500(0.82)^0 = 12500(1) = 12500
Price after one year (f(1)) = 12500(0.82)^1 ≈ 10250
Percentage decrease = [(f(0) - f(1)) / f(0)] * 100
= [(12500 - 10250) / 12500] * 100
= 22%
Therefore, the price of product A is decreasing by approximately 22% 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 for product A with the percentage change for product B.
For product A, we already calculated that the percentage decrease per year is approximately 22%.
To calculate the percentage change for product B, we need to analyze the table provided:
Percentage change from year 1 to year 2:
[(f(1) - f(2)) / f(1)] * 100
[(5600 - 3136) / 5600] * 100 ≈ 43.71%
Percentage change from year 2 to year 3:
[(f(2) - f(3)) / f(2)] * 100
[(3136 - 1756.16) / 3136] * 100 ≈ 43.98%
Percentage change from year 3 to year 4:
[(f(3) - f(4)) / f(3)] * 100
[(1756.16 - 983.45) / 1756.16] * 100 ≈ 43.97%
Comparing the percentage change between product A (22%) and product B (approximately 43.71%, 43.98%, and 43.97%) for each year, we can conclude that product B recorded a greater percentage change in price over the previous year for all three years.
Therefore, product B experienced a greater percentage change in price compared to product A over the previous year.