Question

. An automobile company is ready to introduce a new line of hybrid cars through a national sales campaign. After test marketing the line in a carefully selected city, the marketing research department estimates that sales (in millions of Ghana Cedis) will increase at the monthly rate of S'(t) = 20t − 15e−0.15t 0 ≤ t ≤ 36 t months after the campaign has started. (a) What will be the total sales S(t) in t months after the beginning of the national campaign if we assume no sales at the beginning of the campaign? (b) What are the estimated total sales for the first 12 months of the campaign? (c) When will the estimated total sales reach 100 million Cedis?

Answer 1

We have the rate equation,

We are given,

i.e.,

On rearranging, the above differential equation, we have,

Integrating, both sides of the above equation, from time=0, to time=t, we have,

i.e.,

i.e.,

Now, we know that there is no sales at the beginning of the campaign, therefore,

i.e., , 0 ≤ t ≤ 36

(a) The total sales S(t) in t months after the beginning of the national campaign is gives by,

(b) The estimated total sales for the first 12 months of the campaign, is

i.e.,

i.e.,

i.e.,

i.e., million Cedis

(c) The estimated total sales will reach 100 million Cedis, for the value of t given by,

i.e.,

i.e.,

We can see that if t=4, the LHS of the above equation, will be greater than 20. If t=3, the LHS of the above equation will be less than 20.

Therefore we conclude that the estimated total sales will reach 100 million Cedis, in 4 months.