the exponential model f(t)=a*b^t.

Question

suppose a baseball team's profits are growing according to the exponential model f(t)=a*b^t. Suppose a baseball earns $2 million in 1992 and $18 million in 1994. Find the values of a and b which fit this data

Answer 1

First pick a t = 0

 

Let's say 1992 is t = 0

 

Then;

 

f(0) = ab⁰

 

Anything to the power of zero = 1

 

f(0) = a

 

We know in 1992 f(t) = 2 million $

 

2 = a

 

Now we have our "a" value in millions

 

use t = 2 for 1994 because it is two years after our starting time, 1992.

 

18 = 2 (b²)

 

18/2 = b²

 

9 = b²

 

b = 3

 

So an equation that fits the data is:

 

 

f(t) = 2 * 3^t

 

Where:

t is in years after 1992

f(t) is in millions dollars

Answer 2

f(t) = ab^t

 

Use the given information and solve two equations in two unknowns to find a and b.

 

Assuming t = time elapsed in years from some starting time:

 

Set 1992 as the start time.  Then t = 0 in 1992:

 

2,000,000 = ab^t = ab⁰ = a

 

a = 2,000,000

 

Set 1994 as the t = 2 time (i.e. 2 years after 1992).

 

18,000,000 = 2,000,000 * b²

 

b² = 9

 

b = 3

 

f(t) = 2,000,000 * 3^t with f(t) in $ and t in years elapsed since 1992