Question

Please help
SALES: The rate of increase in sales S (in thousands of units) of a product is proportional to the current level of sales and inversely proportional to the square of the time t. This is described by the differential equation:

dS / dt = kS / t^2

where t is the time in years. The saturation point for the market is 50,000 units. That is, the limit as t approaches infinity is 50. After 1 year, 10,000 units have been sold. Find S as a function of t.

Answer 1

idk im sorry

Answer 2

its all good, i have no idea whats going on

Answer 3

You have:

dS/dt = k*S/(t^2)

As you say, this is a separable equation:

dS/S = k/(t^2) dt

Integrate both sides:

ln(S) - ln(c) = -k/t

where ln(c) is the constant of integration.

ln(S/c) = -k/t

S(t) = c*exp(-k/t)

As t -> ∞, exp(-k/t) -> 1, so

lim(t -> ∞) S(t) = c = 50,000

So:

S(t) = 50,000*exp(-k/t)

We are also told that at t = 1 yr, S(1 yr) = 10,000. We can use this to solve for k:

10,000 = 50,000*exp(-k/yr)

1/5 = exp(-k/yr)

ln(1/5) = -k/yr

-k = ln(1/5)*yr ~= -1.609*yr

So:

S(t) = 50,000*exp(ln(1/5)*yr/t) ~= 50,000*exp((-1.609*yr)/t)

Answer 4

thx alot im gonna examine this

Answer 5

ok