Mathematics

Problem

An economist tracks the Consumer Price Index (CPI), a dimensionless indicator of average consumer prices. On January 1 of a particular year, the CPI is 180. Two years later, the CPI has increased to 198. The economist writes in a report: “The instantaneous rate of change of the CPI is proportional to the current CPI value.” Let

P(t)

denote the CPI

t

years after January 1 of the initial year. a. Translate the economist’s statement into an appropriate differential equation for

P(t)

. b. Determine the proportionality constant

k

by using the CPI data provided for

t=0

and

t=2

. Express

k

as a decimal rounded to four significant figures. c. Using your model, compute the annual percentage rate of inflation implied by the value of

k

. Clearly explain how this rate is obtained. d. Predict the CPI five years after the initial date. Then interpret what this value indicates about overall price changes during the five-year period.

Intuitive explanation

The economist is describing exponential growth, where the rate of change depends on the current amount—a hallmark of differential equations. You'll set up $\frac{dP}{dt} = kP$ using the proportionality statement, then use your two data points ( $P(0) = 180$ and $P(2) = 198$ ) to find the growth constant $k$ and the general solution $P(t) = P_0 e^{kt}$ . This is the key to unlocking real-world phenomena that compound or accelerate on their own.

Intuitive explanation

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