Answer:
A. S(t) = 50 [3 ln t/ln2 +1]
Step-by-step explanation:
We are told that the rate of change in sales S is inversely proportional to time.
Thus;
dS/dt = k/t
Where k is the constant of proportionality.
So,
dS = (k/t)dt
Integrating both sides gives us;
S = (kIn t) + C
We are given that after 2 and 4 weeks are 200 units and 350 units respectively.
Thus;
S(2);
(k In 2) + C = 200 - - - (eq 1)
(k In 4) + C = 350 - - - -(eq 2)
To find k, let's subtract eq 1 from eq 2 to get;
(k In 4) - (k In 2) = 350 - 200
(k In 4) - (k In 2) = 150
k(In 4 - In 2) = 150
0.6931k = 150
k = 150/0.6931
k = 216.42
Plugging this for k in eq 1 gives;
(216.42 In 2) + C = 200
C = 200 - 150
C = 50
Thus;
S(t) = (216.42In t) + 50
Now 216.42 can also be expressed as;
150/In 2
Thus;
S(t) = ((150/In 2)In t) + 50
Factorizing out gives;
S(t) = 50 [(3 ln t/ln2) + 1]
Option A is the correct answer
