The total revenue earned by a clothing store chain at time t is given by R(t) = N(t)S(t), where N(t) is the number of stores at time t and S(t) is the average revenue per store per month at time t. The initial amount of stores the company operates is 50 and the initial average revenue per store per month is $150, 000.
(a) Use the product rule to determine the change in the total revenue earned at time t.
(b) If the company builds stores at a rate of 5 stores per month, but expects average revenue per store per
month to stay constant, what is the expected change in the total revenue when t = 0?
(c) If the company decides to use advertising to increase average revenue per store at a rate of $10,000 per
month, but keeps the number of store constant, what is the expected change in the total revenue when
t = 0.
(d) Suppose the company decides to both increase the number of stores and use advertising to increase average revenue per store per month (at the same rates as in parts (b) and (c)). What is the expected change in the total revenue when t = 0?
dR/dt = N dS/dt + S dN/dt
Ni = 50
Si = 150,000
initial R = N S = 7,500,000
so
dR/dt = 50 dS/dt + 150,000 dN/dt
b. dN/dt = 5 and dS/dt = 0
dR/dt = 0 + 750,000 dollars per month
c. dN/dt = 0 and dS/dt = 10,000
dR /dt = 50 * 10,000 +0 = 500,000 dollars/month
d. add them :)