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)
I've been stuck on this all day :-(
I think you have a very fundamental typo.
We are talking about a constant rate of inflation here
I bet you mean
f(x) = 0.69 (1.03)^x
in other words TO THE POWER x
that would be 0.69 when x = 0 years, now
0.69 * 1.03 after one year
0.69 * 1.03 *1.03 after two years
0.69 *1.03*1.03*1.03 after 3 etc
It is getting bigger. Every year you increase by .03 x which is 3%
In your table in part B
the ratio is for example
10406.04/10303.01
= 1.01
check:
10303.01/10201 = 1.01 sure enough
so that is increasing by ONLY ONE percent per year
thank you so much this has helped me so much because i was stuck on this question for some time.
I can help you with that! Let's start by answering Part A.
To determine whether the price of product A is increasing or decreasing, we need to examine the function f(x) = 0.69(1.03)^x.
In this function, the base number 1.03 represents the rate of change per year. If the base number is greater than 1, it indicates an increase over time, and if it's less than 1, it indicates a decrease.
In this case, the base number 1.03 is greater than 1, which means the price of product A is increasing over time. Now let's find the percentage increase per year.
We can calculate the percentage increase per year by subtracting 1 from the base number and then multiplying by 100.
Percentage increase per year = (1.03 - 1) * 100 = 0.03 * 100 = 3%.
Therefore, the price of product A is increasing by 3% per year.
Now, let's move on to Part B and compare the percentage change in price for products A and B.
We have the following price values for product B after different numbers of years:
t (number of years) | f(t) (price in dollars)
1 | 10,100
2 | 10,201
3 | 10,303.01
4 | 10,406.04
To find the percentage change in price over the previous year for product B, we need to compare each year's price with the price of the previous year.
For example, to find the percentage change from year 1 to year 2, we can use the following formula:
Percentage change = ((f(2) - f(1)) / f(1)) * 100
Applying this formula to each consecutive pair of years, we get:
Percentage change from year 1 to year 2 = ((10,201 - 10,100) / 10,100) * 100
Percentage change from year 2 to year 3 = ((10,303.01 - 10,201) / 10,201) * 100
Percentage change from year 3 to year 4 = ((10,406.04 - 10,303.01) / 10,303.01) * 100
By calculating these percentages, you can compare the percentage change in price over the previous year for products A and B.
After comparing the percentages, you can determine which product recorded a greater percentage change in price over the previous year.
I hope this explanation helps! Let me know if you have any further questions.