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.

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

jepessoa jepessoa · Answer 2 · 2020-04-02T18:18:28+02:00

Answer:

$354.42

Explanation:

First we must determine the exponential growth rate:

y = aeⁿˣ

a = 1967 value = $100
n = time = 10
y = 1977 value = $148.50
x = exponential growth rate

148.5 = 100e¹⁰ˣ

e¹⁰ˣ = 148.5 / 100 = 1.485

ln (e¹⁰ˣ) = ln 1.485

10x = 0.39541

x = 0.039541

Now we replace x into a new equation:

y = aeⁿˣ

a = 1967 value = $100
n = time = 32 (1999 - 1967)
y = 1999 value = ?
x = exponential growth rate = 0.039541
e = given mathematical number = 2.71828

y = 100 (2.71828⁽³²⁾⁽⁰°⁰³⁹⁵⁴¹⁾) = 100 (2.71828¹°²⁶⁵³¹²) = 100 (3.5442)

y = 354.42