Question

Given the inflation model c(t)=c₀(1−r)ᵗ and given that a car that currently sells for $31,400 sold for $15,700.08 years ago, what has the average annual rate of inflation been for the past eight years?

a. 5%
b. 7%
c. 10%
d. 15%

Answer 1

The average annual rate of inflation over the past eight years is 10%, determined by solving the inflation model equation c(t)=c₀(1−r)ᵗ, given that a car's price changed from $15,700 to $31,400 in that period.

Step-by-step explanation:

To calculate the average annual rate of inflation over the past eight years using the given inflation model c(t) = c₀(1−r)ᵗ, we need to figure out the constant rate r that fits the data about car prices provided. The initial price of the car was $15,700 eight years ago, and it is now $31,400. We can substitute these values into the inflation model to find r.

Let c₀ be the initial price (eight years ago) and c(t) be the current price ($31,400). Thus, we get the equation $15,700(1 − r)⁸ = $31,400.

Dividing both sides by $15,700 gives us (1 − r)⁸ = 2. After, we take the eighth root of both sides, giving 1 − r = √[⁸]{2}. From here, we solve for r and find that the rate r is approximately 0.10 or 10%. Hence, the annual rate of inflation over the past eight years is 10%, which answers the student's question with option (c).

The impact of inflation is significant in economics as it can drastically affect the purchasing power of consumers. Inflation rates can vary greatly, from low inflation to the extreme cases of hyperinflation. Understanding these rates is crucial for financial planning and economic analysis.