Question

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

A. 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.

Answer 1

Lets look at product B first. How much is it increasing each year?
Look at 10,201 / 10,100. What do you get?

Answer 2

1.01

Answer 3

Same for 10303.1 / 10201 so that's the common ratio?

Answer 4

Now what?

Answer 5

you have 4 years values for product B

you may also want to get 4 years values table for A too, so you can compare the percentage differences

Answer 6

\(\large \begin{array}{ccccclll}
\textit{Product A}\\
years&1&2&3&4 \\
price&0.7107&1.4214&2.1321&2.8428\\
\hline\\
\textit{Product B}\\
years&1&2&3&4\\
price&10,100&10,201&10,303.01&10,406.04\\


\end{array}\)

Answer 7

so for example, to know the percentages from year to year, you calculate the difference between, in percentage terms

lemme do product B for the 2nd and 3rd years, in relation to their previous one

so 1st year is 10,100, 2nd year is 10,201 so it went up in price by 201, how much is that in percentage from 10,100 <-- previous year

well \(\large {\begin{array}{llll}
quantity& percentage\\
10,100&100\\
201&x


\end{array}\implies \cfrac{10,100}{201} = \cfrac{100}{x}\\ \quad \\
x = \cfrac{201\times 100}{10,100} \approx 1.99\%}\)

Answer 8

Thank you so much!

Answer 9

so now let's see again Product B for the 3rd year

so 3rd year is 10,303.01, the previous year, 2nd year, was 10,201, so it went up by 102.01

well then \(\large {\begin{array}{lllllll}
quantity& percentage\\
10,201&100\\
102.01&x


\end{array}\\ \quad \\\implies \cfrac{10,201}{102.01} = \cfrac{100}{x}\implies x = \cfrac{102.01\times 100}{10,201} \implies 1\%}\)

Answer 10

and so on for all prices, so you'd end up with the percentages for each year.... the 1st year has no previous so, you'd start with the 2nd and 3rd and 4th year

then you can compare the percentages of them, see how they changed from one year to the next