A craft distillery bottles a line of artisanal soda
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A craft distillery bottles a line of artisanal soda. The market price (in dollars) required to sell q bottles per day is modeled by p(q)=18−0.05q, where 0≤q≤360. The variable production cost is 6 dollars per bottle, and fixed operating costs are 300 dollars per day. In addition, beginning with the 200th bottle produced each day, an excise tax of 1.50 dollars per bottle is imposed; the tax does not apply to the first 199 bottles.
(a) Express the daily cost C(q) (in dollars) as a piecewise function of q that accounts for fixed cost, variable production cost, and the excise tax.
(b) Write an explicit piecewise expression for the daily profit P(q) in terms of q. State the domain on which P is defined.
(c) Determine the production quantity (to the nearest whole bottle) that maximizes daily profit. Be sure to consider the effect of the tax discontinuity at q=200 in your analysis and show complete calculus justification.
(d) For the optimal production level obtained in part (c), compute the selling price per bottle and the corresponding maximum daily profit, rounding monetary answers to the nearest cent.