exponential decay

Question

The sales, s, of a product have declined in recent years. there were 202 million sold in 1984 and 1.4 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

Initially, we are dealing with a time frame of 10 years from t=0

in 1984 to t = 10 in 1994.  Our final sales S = 1.4 million, and our

initial sales S₀ = 202 million

 

1,400,000 = 202,000,000e^-10k

7/1010 = e^-10k

ln(7/1010) = ln(e^-10k)

-10k = ln(7/1010)

k = ln(7/1010)/(-10)

k = .49718

 

This means that the generic equation for sales as a function of time is:

S(t) = Se^-0.49718t

 

You can check this by verifiying with the numbers initially given.

If our formula is correct then

202000000e^-.49718(10) should be approximately 1.4 million

 

202,000,000e^-4.9718 = 1,399,994

I would say this is within the acceptable parameters.

 

 

 

Answer 2

a) We know that at t = 0 (year 1984) we have S₀ = 202 (in millions). Also, S(10) = 1.4 (in millions). So then the equation:
S = S₀*e^-kt;

=> S(10)/S₀ = e^(-k*10)  (use the known sales)

=> -10*k = ln(1.4/202)  (take the natural log of both sides)

=> k = (1/10)*ln(202/1.4)  (divide by -10)

=> k = 0.497.

 

b) t = 2002 - 1984 = 18 years.

S(18) = S₀*e^(-k*18)   (plug in t = 18)

=> S(18) = 202*e^(-0.497*18)   (substitute S₀ and k)

=> S(18) = 0.0263 million sales   (evaluate)

=> S(18) = 26,312 sales.    (rewrite units)

 

c) S(t) = 1 sale. S₀ = 202,000,000 sales

1 = 202,000,000*e^(-0.497*t)   (Substitute values)

=> -0.497*t = ln(1/202,000,000)   (Take the natural log of both sides)

=> t = (1/0.497)*ln(202,000,000)     (Solve for time)

=> t = 38.5 years   

year = t + 1984 = 2022.  (Find the year by using that when t = 0, year is 1984).

Halfway through year 2022 (so theoretically year 2023)