A craft-beverage company sells specialty cans of sparkling water
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A craft-beverage company sells specialty cans of sparkling water. The pre-tax demand price for selling q thousand cans per month is p(q)=4.50−0.02q, in dollars per can. The canning plant's monthly cost is modeled by C(q)=0.3q2+1.2q+8, where C is measured in thousands of dollars. A new environmental tax of 0.15 per can is imposed on each unit sold. The company must pay this tax directly to the government for every can it produces. Let P(q) be the company's monthly profit, in thousands of dollars, after the tax is imposed.
(a) Express the company's post-tax revenue function Rt(q) in terms of q. Clearly incorporate the tax into your model.
(b) Write a formula for the profit function P(q) in terms of q.
(c) Compute P′(50) and interpret its numerical value, including proper units.
(d) Based on your calculation in part (c), determine whether increasing production beyond 50 thousand cans per month will immediately increase or decrease profit. Explain your reasoning without using a graph.