3 Part Word Problem Pre Calc

Question

The sales, s, of a product have declined in recent years. there were 202 million sold in 1984 and 1.2 million sold in 1994. Assume the sales are decreasing according to the exponential decay model, S(t)=Se^-kt.

a) Find the value k and write an exponential f(x) that describes the number sold after time, t, in years since 1984

b)Estimate the sales of the product in the year 2002

c) In what year (theoretically) will only 1 of the product be sold?

Answer 1

'84 202 million

'94 1.2 million

1.2 = 202e^10t

ln(1.2/202) = 10k

k = ln(.6/101)/10 = about -.5126

= about 51% annual decay rate

in '02, t= 2002-1984 = 18

A = 202e^-.5126(18)= 202/e^.5126(18)

= about 0.0198721633 million

= about 19,872 to be sold in 2002

.000001=202/e^.5126t

1 =0.000001 million

.000001/202 = e^-.5126t

ln(.00001/202) = -.5126t

t = ln(.000001/202)/-.5126

= about 37.3074

year = 1984+ 37 .3

= about the year 2021 or slightly into 2022 when sales go to just 1 item sold