a) find the value k and write an exponential function that describes the number sold after time, t, in years since 1984.
For this case we have an equation of the form:
[tex] s (t) = s0 * e ^ {-kt}
[/tex]
From here, we must find the values of s0 and k.
For this, we use the following data:
There were 201 million sold in 1984 and 1.3 million sold in 1994.
Therefore, the initial sales are:
[tex] s0 = 201 million
[/tex]
Then, the value of k is given by:
[tex] 1,300,000 = 201,000,000e ^ {-10k} [/tex]
Clearing k we have:
[tex] (1,300,000) / (201,000,000) = e ^ {-10k}
0.006467662 = e ^ {-10k}
ln (0.006467662) = ln (e ^ {-10k})
[/tex]
[tex] -10k = -5.040940596
k = (-5.040940596) / (- 10)
k = 0.5041
[/tex]
Thus, the generic equation is:
[tex] s (t) = (201000000) * e ^ {- (0.5041) t}
[/tex]
b) estimate the sales of the product in the year 2002
For the year 2002 we have:
[tex] s (18) = (201000000) * e ^ {- (0.5041) (18)}
s (18) = 23041
[/tex]
c) in what year (theoretically) will only 1 of the product be sold?
By the time 1 single product is sold, we have:
[tex] 1 = (201000000) * e ^ {- (0.5041) t}
[/tex]
Clearing the time we have:
[tex] (1) / (201,000,000) = e ^ {-0.5041t}
4.98 * 10 ^ {-9} = e ^ {-0.5041t}
ln (4.98 * 10 ^ {-9}) = ln (e ^ {-0.5041t})
[/tex]
[tex] -0.5041t = -19.11783595
t = (-19.11783595) / (- 0.5041)
t = 38 years
[/tex]
Therefore, only 1 product will be sold after 38 years.
