Question

1950, the U.S. federal budget was $39.4 billion, in 2000, the federal budget was $2025.2 billion. Find an exponential function to model this data

Answer 1

Let's do the usual thing and make t the years since 1950. We'll just abbreviate a billion B.

f(1950-1950)=39.4 B

f(2000-1950) =2025.2 B

Our exponential form for f will be

`f = a e^(kt)`

`39.4 \textrm{ B} = a e^( 0 k) = a`

`2025.2 \textrm{ B} = a e^(50 k)`

Dividing

`(2025.2)/(39.4) = e^(50 k)`

`50 k = \ln (2025.2)/(39.4)`

`k = \frac 1 {50} \ln (2025.2)/(39.4) \approx 0.0787932`

Our function is

`f = 39.4 \textrm{ B } e^(0.0787932 t )`

Since
`e^(0.0787932) \approx 1.08198`

`f = 39.4 \textrm{B } 1.08198^t }`

around 8.2 % annualized growth.