Question

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

Answer 1

S = 50000/5^(1/t)

Explanation:

Since we have that 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.

And ds/dt means the rate of increase in S

dS/dt ∝ S/t².

Now,

dS/dt = kS/t². Where k is the constant of proportionality

Solving this equation,

dS/S = kdt/t²

Taking the integral of both sides,

∫ dS/S = ∫ kdt/t²

Integrating both sides, we get,

lnS = - k/t + C

As t --> ∞,

ln50000 = C.

So,

lnS = - k/t + ln50000.

Next,

When t = 1, S = 10000.

We have,

ln10000 = - k + ln50000.

k = ln50000 - ln10000.

k = ln5.

Now,

lnS = - 1/t. ln5 + ln50000.

Taking the exponential of both sides, we get,

S = 50000/5^(1/t)