Question

Suppose that a fruit juice manufacturer determines that its weekly cost, in dollars, of producing x thousands of bottles of apple juice is given by C(x)-285+0.5x+0.01x In the market, the demand function for that brand of apple juice is found to be x-D(p)-2000-400p , where p is in dollars. (a) Rewrite the demand function to express p as a function of x : P=D(x)= (b) The revenue function is R(x)- (c) For the company to make the weekly maximum profit, produced and sold each week. thousands of bottles of apple juice ovens should be

Answer 1

The student's question involves rewriting a demand function to express price as a function of quantity, deriving a revenue function, and determining the quantity that maximizes profit for a juice manufacturer.

Step-by-step explanation:

The student is dealing with a question in economics, specifically involving the determination of a revenue function and the maximization of profit for a manufacturer of apple juice. To address part (a) of the question, the demand function, which is initially given as a function of price (p), must be re-written as a function of quantity (x). This inverse demand function will express price as a function of quantity, P = D(x).

To find the revenue function in part (b), we need to know that revenue (R) is calculated as the product of the price (p) and the quantity (x). Since we have p as a function of x from part (a), we can compute R(x) = x * P(x).

Finally, for part (c), the company aims to maximize profit, which is the difference between revenue and cost. This typically involves finding the derivative of the profit function and setting it equal to zero to find critical points, and then using the second derivative test or other methods to find the maximum.