A small notebook manufacturer has the following cost and revenue structure for a…
MATHEMATICS · Multi-part · AoPS · Mathematics · Special Functions · Special Functions
Problem
A small notebook manufacturer has the following cost and revenue structure for a single working day:
• Fixed overheads: Rs. 20,000 irrespective of the number of notebooks produced.
• Variable manufacturing cost: Rs. 25 per notebook for the first 1,000 notebooks; Rs. 20 per notebook for every notebook produced beyond 1,000.
• Selling price: Rs. 40 per notebook, assumed to remain constant for all units sold.
Let x(≥0) denote the number of notebooks produced and sold in a day. Define the daily profit Π(x) (in rupees) as revenue minus total cost.
(i) Write Π(x) as a piecewise-defined function of x, clearly indicating the rule applicable in each part of the domain.
(ii) Show that Π is a function on the set of non-negative integers and explain why each value of x gives a unique profit figure.
(iii) For 0≤x≤3,000, determine the interval(s) of x for which the manufacturer earns a non-negative profit.
(iv) Identify the break-even point(s) and interpret them in this situation.
(v) State the feasible domain and the corresponding range of Π for the production cap of 3,000 notebooks. Comment on the nature of the graph in view of the piecewise definition.