# The consumer price index (CPI) for a given year is the amount of money in that year that has the...

## Question:

The consumer price index (CPI) for a given year is the amount of money in that year that has the same purchasing power as \$100 in 1983. At the start of 2009, the CPI was 211. Write a formula for the CPI as a function of t , years after 2009, assuming that the CPI increases by 2.7% every year.

## Constructing an Exponential Function

Depending on how a quantity is defined, we may be able to model it with an exponential function. If we know that the quantity increases by an amount defined by a constant percentage, then we have an exponential function. Assuming this percent is a rate r, we can define the function as follows.

{eq}f(t) = A(1+r)^t {/eq}

Since the CPI is growing by 2.7% each year, we know that we need to model this using an exponential function. As we have both this rate and the initial value in 2009, we can define this function using those values. The initial value is the coefficient in front of the exponential base, and the percent is being added to 1 as the exponential base.

{eq}C(t) = 211(1+0.027)^t = 211(1.027)^t {/eq}