Application of Logistic Growth Word Problem

Question

After a new product is launched the cumulative sales S(t) (in $1,000) t weeks after launch is given by:

S(t)=72/(1+9e^-0.36t)

a. Determine the cumulative amount in sales 3 weeks after launch. Round to the nearest thousandth.

b. Determine the amount of time required for the cumulative sales to reach $70,000.

c. What is the limiting value in sales?

Answer 1

S(t) = 72/(1+9e^-0.36t)

a) Just plug in t = 3 and compute the answer. Use your calculator.

b) S(t) = $70 thousand

70 = 72/(1+9e^-0.36t)

1 + 9·e^-0.36t = 72/70

9·e^-0.36t = 72/70 - 1 = 0.0286 (rounded to three decimal places)

e^-0.36t = 0.0286/9 = 0.003

ln(e^-0.36t) = ln(0.003)

-0.36t = ln(0.003)

Solve for t, Use your calculator to get the answer. The units are weeks.

c) As t → ∞, e^-0.36t → 0; so S(t) → 72/(1+0) → $___?