Question

Please help!

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) = 72(1.25)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)
65 84.5 109.85 142.81

Which product recorded a greater percentage change in price over the previous year? Justify your answer. (5 points)

Answer 1

Explanation:

A

f(x) = 72 × 1.25 × x

so, the price is $90 after 1 year (72×1.25×1).

$180 after 2 years (72×1.25×2)

$270 after 3 years (72×1.25×3)

and so on.

or did you mean

f(x) = 72 × (1.25)^x

so, still, the price is $90 after 1 year (72×1.25^1).

$112.50 after 2 years (72×1.25^2).

$140.625 = $140.63 after 3 years (72×1.25^3)

that makes more sense. I will continue with that assumption.

so, the price is increasing, as every factor of the multiplication is positive and larger than 1. therefore, it only goes up.

and due to the formula, the price of every year is the price of the previous year multiplied by 1.25.

a multiplication by 1.25 is an increase of 25% (compared to a constant multiplication by 1).

so, the price increases every year by 25%.

B

year 1

price $65

year 2

price $84.50 difference $19.50

100% = 65

1% = 65/100 = 0.65

19.5/0.65 = 30% difference

year 3

price $109.85 difference $25.35

100% = 84.5

1% = 84.5/100 = 0.845

25.35/0.845 = 30% difference

year 4

price $142.81 difference $32.96

100% = 109.85

1% = 109.85/100 = 1.0985

32.96/1.0985 = 30% difference

so, product B has the greater percentage increase in the price (30% vs. 25%).

Answer 2

A)25% B)Product B

Explanation:

A) To find whether product A price is increasing or decreasing, we will find the difference in price between year 0 and 1 year after.

When x = 0,

f(0)
`72(1.25)^(0)\\= 72*1\\= 72`

when x = 1,

f(1) =
`72(1.25)^(1)\\= 72*1.25\\= 90`

From here we can already see the price is increasing.

% increase = (Difference in price / Original Price) * 100

=
`(90-72)/(72) *100\\=(18)/(72) *100\\=0.25*100\\=25percent`

Therefore the price is increasing by 25% per year.

B) Same concept as Part A, we will find the price difference between year 1 and 2.

% increase = (Difference in price / Original Price) * 100

`(84.5-65)/(65) * 100\\= (19.5)/(65) * 100\\= 0.3*100\\= 30 percent`

Therefore we can see here Product B has a higher increase than Product A.