Question

The price of fuel may increase due to demand and decrease due to overproduction. Marco is studying the change in the price of two types of fuel, A and B, over time.
The price f(x), in dollars, of fuel A after x months is represented by the function below:

f(x) = 2.15(0.98)x

Part A: Is the price of fuel A increasing or decreasing and by what percentage per month? Justify your answer. (5 points)

Part B: The table below shows the price g(m), in dollars, of fuel B after m months.

m (number of months) 1 2 3 4
g(m) (price in dollars) 4.19 3.98 3.78 3.59
Which type of fuel recorded a greater percentage change in price over the previous month? Justify your answer. (5 points)

(10 points)

Answer 1

Answer:

Part A) The price of fuel A is decreasing by a percentage of 2% per month

Part B) Fuel B recorded a greater percentage change price over the previous month (5%) than the Fuel A (2%)

Step-by-step explanation:

Part A: Is the price of fuel A increasing or decreasing and by what percentage per month?

we know that

The equation of the exponential function is equal to

$f(x)=a(1+r)^x$

where

x ---> the number of months

f(x) ----> is the price in dollars

a ----> is the initial value or y-intercept

r ---> is the percent rate of change

we have

$f(x)=2.15(0.98)^x$

so

$a=\ 2.15\\(1+r)=0.98$

Find the value of r

$r=0.98-1=-0.02=-2\%$

The rate of change is negative

That means ----> Is a exponential decay function

therefore

The price of fuel A is decreasing by a percentage of 2% per month

Part B) The table below shows the price g(m), in dollars, of fuel B after m months.

we know that

The exponential function is equal to

$g(m)=a(1+r)^m$

where

m ---> the number of months

g(m) ----> is the price in dollars

a ----> is the initial value or y-intercept

r ---> is the percent rate of change

Find the value of r

For m=1, g(m)=4.19

substitute

$4.19=a(1+r)^1$

$4.19=a(1+r)$ ----> equation A

For m=2, g(m)=3.98

substitute

$3.98=a(1+r)^2$ ----> equation B

Divide 3.98 by 4.19

$\frac{3.98}{4.19}=\frac{a(1+r)^2}{a(1+r)}$

simplify

$0.95=1+r\\r=0.95-1\\r=-0.05=-5\%$

Is negative because is a decreasing function

so

The price of fuel B is decreasing by a percentage of 5% per month

therefore

Fuel B recorded a greater percentage change price over the previous month (5%) than the Fuel A (2%)