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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. (5 points)

Someone please help !!! :(



Answer :

semsee45

Answer:

A)  Decreasing by 18% per year.

B)  Product A recorded a greater percentage change in price over the previous year.

Step-by-step explanation:

Exponential Function

[tex]\large\boxed{y=ab^x}[/tex]

where:

  • a is the initial value (y-intercept).
  • b is the base (growth/decay factor) in decimal form.
  • x is the independent variable.
  • y is the dependent variable.

If b > 1 then it is an increasing function.

If 0 < b < 1 then it is a decreasing function.  

Part A

Given function:

[tex]f(x)=12500(0.82)^x[/tex]

Therefore:

  • a = 12500
  • b = 0.82

As the base of the given exponential function is greater than zero but less than 1, it is a decreasing function.

The percentage the function decreases by per year is:

[tex]\implies 1-b=1-0.82=0.18=18\%[/tex]

Part B

Given table for Product B:

[tex]\begin{array}{|c|c|c|c|c|}\cline{1-5} t & 1 &2 &3 &4\\\cline{1-5} f(t) &5600 &3136& 1756.16 &983.45\\\cline{1-5}\end{array}[/tex]

where:

  • f(t) = price of product B (in dollars).
  • t = time (in years).

Therefore, an exponential equation modelling the change in price is:

[tex]f(t)=ab^t[/tex]

Substitute two of the ordered pairs into the exponential formula:

[tex]\implies 5600=ab^1[/tex]

[tex]\implies 3136=ab^2[/tex]

Divide the second equation by the first equation to eliminate a and solve for b:

[tex]\implies \dfrac{ab^2}{ab}=\dfrac{3136}{5600}[/tex]

[tex]\implies b=0.56[/tex]

Therefore, the base an exponential function modelling the price of Product A is 0.56.  This means that the Product A decreases by 44% per year since:

[tex]\implies 1-b=1-0.56=0.44=44\%[/tex]

Therefore, the product that recorded a greater percentage change in price over the previous year is Product A as 44% > 18%.

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