An online retailer sells two related styles of wireless earbuds, Model A and Model B
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
An online retailer sells two related styles of wireless earbuds, Model A and Model B. Let qA and qB denote the daily sales, in units, of Models A and B, respectively. Market research suggests the following price-demand relationships:
pA=85−0.10qA−0.05qB,pB=75−0.08qB−0.04qA,
where pA and pB are the selling prices in dollars per unit. The variable cost of each unit of Model A is 37, and that of each unit of Model B is 31. Fixed daily operating costs are 6000. Let R(qA,qB) be the total daily revenue and P(qA,qB) be the total daily profit.
(a) Write an explicit formula for the total revenue R(qA,qB) in terms of qA and qB. Do not combine like terms.
(b) Obtain expressions for the marginal profits ∂qA∂P and ∂qB∂P.
(c) The firm currently sells 200 units of each model daily. Evaluate both marginal profits at (qA,qB)=(200,200), and interpret their numerical values in practical terms.
(d) Suppose the company considers issuing a flash discount that will immediately increase Model A sales at a rate of 15 additional units per day while leaving Model B sales unchanged. Use a total differential to estimate the instantaneous rate of change of total profit at the moment the discount begins. State whether profit is increasing or decreasing and provide a brief justification.