The consumer price index compares the cost of goods and services over various years. The same goods and services that cost $100 in 1967 cost $148.50 in 1977. Assuming exponential growth, find the value of these same goods and services in 1999.
Business · High School · Tue Nov 03 2020
Answered on
Answer:
The cost of the goods and services is approximately $970.05 in the year 1999
Explanation:
In this question, we are asked to calculate the cost of a particular goods and services in the year 1999, which cost a certain amount in the year 1967 and 1997.
We proceed as follows;
$100 = base year (1967)
$148.50 in 1977, which is "Year 10" because 1977-1967 = 10
$ ? in 1999, which is "Year 32"
Exponential growth generally looks like:
y = Pe^rt where P = initial value, r = "rate," and t = time
We know the following ordered pairs:
(0, 100) and (10, 148.50)
Plugging them in...
100 = Pe^r(0) = P so P = 100
Now let's do the same with the next ordered pair
148.50 = (100)(e^10r)
148.50/100 = e ^10r
1.485 = e^10r
ln(1.485) = 10r
0.1ln(1.485) = r
Now we plug that in for our r to get the formula
Y = (100)(e^ (0.1ln(1.485) t)
Now we plug in year 32 and solve for y
Y = (100)(e^ (0.1ln(1.485) t) =
Y = (100)(e^ 3.2ln(1.485)
Y = approx. $970.05 in 1999