Question

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) = 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)

Answer 1

Answer:

Product A has a greater percentage change in price.

Step-by-step explanation:

Part A:

The price f product A, f (x) after x years is given by: 

 $f(x) = 0.69\cdot(1.03)^{x}$

After x = 0 years, the price of product A is:

$f(0) = 0.69\cdot(1.03)^{0}=0.69$

After x = 1 years, the price of product A is:

$f(1) = 0.69\cdot(1.03)^{1}=0.69\cdot (1+0.03)=0.69\cdot (1+3\%)$

After 1 year, the price of product A is 3% times more than the original price.

This means that after one year, the new price is 103% of the original price, which means the price product A is increasing by 3%.

Again after x = 2 years, the price of product A is:

$f(2) = 0.69\cdot(1.03)^{2}=[0.69\cdot (1+3\%)]\times (1.03)$

This implies that after 2 years, the price of product A is 103% of the price after year 1.

This implies that the price of product A is 3% more than the previous year.

Thus, the price of product A is increasing each year by 3%.

Part B:

The data for Product B is as follows:

Time (t)          Price [f (t)]

   1                   10,100

   2                   10,201

   3                 10,303.01

   4                 10,406.04

Product B is clearly increasing in price.

Consider the changes in price of Product B in the following intervals of years:

• Year 1 - Year 2:

Price in year 1 = $10,100

Price in Year 2 = $10,201

Compute the increase percentage as follows:

$\text{Increase}\%=\frac{10201-10100}{10100}=0.01=1\%$

• Year 2 - Year 3:

Price in Year 2 = $10,201

Price in year 3 = $10,303.01

Compute the increase percentage as follows:

$\text{Increase}\%=\frac{10303.01-10201}{10201}=0.01=1\%$

• Year 3 - Year 4

Price in year 3 = $10,303.01

Price in Year 4 = $10,406.04

Compute the increase percentage as follows:

$\text{Increase}\%=\frac{10,406.04-10303.01}{10303.01}=0.09999\approx 0.01=1\%$

It is quite clear that the price of product B increases by 1% each year.

Thus, Product A has a greater percentage change in price.