Question

The revenues R (in millions of dollars) of skiing facilities in the United States from 2000 through 2008 can be modeled by

R = 1548e0.0617t
where t = 0 corresponds to 2000.
(a) Use this model to estimate the revenues in 2006.
(b) At what rate were the revenues changing in 2006?

Answer 1

a) The revenues in 2006 were of 2.24 billion dollars.

b) The revenues increased 138.3 million dollars in 2006.

Explanation:

We have the following expression to model the revenues in t years after 2000.

`R(t) = 1548e^(0.0617t)`

The revenue is calculated in millions of dollars.

(a) Use this model to estimate the revenues in 2006.

2006 is 6 years after 2006.

So this is R(6)

`R(t) = 1548e^(0.0617t)`

`R(6) = 1548e^(0.0617*6) = 2241.54`

The revenues in 2006 were 2241.54 millions of dollars, or 2.24 billion dollars.

(b) At what rate were the revenues changing in 2006?

The rate is the derivative. So this is R'(6).

`R(t) = 1548e^(0.0617t)`

`R'(t) = 0.0617*1548e^(0.0617t)`

`R'(t) = 95.51e^(0.0617t)`

`R'(6) = 95.51e^(0.0617*6) = 138.3`

The revenues increased 138.3 million dollars in 2006.