Question

1. Let S(t) denotes the value of sales of a particular commodity per unit of time, evaluated at time t. In a stable market where no sales promotion is carried out, the decrease in S(t) per unit of time is proportional to S(t).Thus sales decelerate at the constant proportional rate a>0, implying that Ṡ=-aS(t)
   1. Find the expression for S(t) when sales at time 0 are S₀.
   2. Solve the equation S₀e^-at=0.5S₀ .

Answer 1

Part a)

we know that:

Where = dS(t)/dt

Hence:

dS(t)/dt = -aS(t)

dS(t)/S(t) = -a dt

Integrating on both sides from 0 to t we get:

[ ln(S(t)) ]t=0t=t   = -a (t)0t

Where the subscript is the lower limit and the superscript is the upper limit.

Hence we get:

ln(S(t)) - ln(S(0)) = -at

ln( S(t)/S(0) ) = -at

Hence taking the exponent on both sides we get:

S(t)/S(0) = e-at

Hence:

S(t) = S(0) * e-at

Part b)

S(0) * e-at = 0.5 * S(0)

Hence:

e-at = 0.5

Hence taking natural log on both sides we get:

-at = ln(0.5) = ln(1/2) = -ln(2)

Hence:

-at = -ln(2)

Hence:

at = ln(2)

Hence:

t = ln(2)/a

t = 0.6931/a