Question

After a new product is launched the cumulative sales S(t) (in $1000) t weeks after launch is given by:

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

Required:
a. Determine the cumulative amount in sales 3 weeks after launch.
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

$17.750 ; 15.979 ; 72

Explanation:

Given that :

Cummulative sales, S(t) is represented by the equation :

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

Cummulative sales after 3 weeks :

Put t = 3 in the equation, as t = time after launch

S(3) = 72/(1 + 9e^-0.36(3))

S(3) = 72 / (1 + 9e^-1.08)

S(3) = 72 / (1 +3.0563597)

S(3) = 72 / 4.0563597

S(3) = 17.749905 = $17.750 thousands

Amount of time required for sales to reach 70000

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

S(t) = 70

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

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

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

1 + 9e^-0.36t = 1.0285714

9e^-0.36t = 1.0285714 - 1

9e^-0.36t = 0.0285714

e^-0.36t = 0.0285714 / 9

e^-0.36t = 0.0031746

Take the In of both sides ;

In(e^-0.36t) = In(0.0031746)

-0.36t = - 5.752573

t = - 5.752573 / - 0.36

t = 15.979

About 16 weeks

The limiting value in sales :

Take the limit as t - - > ∞

S(t - - > ∞) = 72/(1 + 9e^-0.36t)

Put t = 0

S(0) - - > 72 / (1 + 0)

72 / 1

= 72