Calculus Help - Demand Relation, Revenue, Cost and Profit Functions?

Question

I'm in an applied calculus course and my instructor assigned a set of problems basically designed to make us brainstorm and put together multiple aspects of calculus to form our solutions. Any help you can provide on this problem would be greatly appreciated.

Phillip, the proprietor of a vineyard, has figured out that he could sell 10,000 bottles of wine at a price of $15 per bottle. Furthermore that for every $0.10 increase in the bottle price, 500 fewer bottles are sold. Phillip also calculated that the cost to produce a bottle of wine is $3 per bottle, plus the fixed costs of $2500.

(a) Find the demand relation, revenue, cost and profit functions for Phillip.
(b) What price should Phillip price the bottles of wine to maximize his revenue?
(c) What price should Phillip price the bottles of wine to maximize his profit?

I just wanted to thank both of you so far who answered my question. I knew what the question is asking, I was just unsure of how to mold the information given into what I needed to answer the problem. But both of you helped immensely to guide me towards the correct solution.

Answer 1

Does the information that a $.10 increase in price resulting in a reduction in demand by 500 also mean a $.10 decrease in price will result in an increase in demand by 500? if so,

let y be the demand, x be the price

y = 10000 - 500*((x-15)/.1) = 10000 - (500/.1 * (x-15))
= 10000 - 5000x + 75000 = 85000 - 5000x

revenue = demand * price = (85000 - 5000x) * x = 85000x - 5000x^2
cost = 2500 + 3x
profit = revenue - cost = 85000x - 5000x^2 - (2500 + 3x)
= 84997x - 5000x*2 -2500

Max revenue will occur when the derivative of the revenue function is zero, i.e 85000 - 10000x = 0 , which is a price of $8.50, which results in a demand of 42500 bottles and a revenue of $361,250 .

Max profit will occur when the derivative of the profit function is zero, i.e. 84997 - 10000x = 0, which is a price of $8.4997 which results in a demand of 42,501.5 bottles, revenue of $361,249.99955, cost of $2525.4991 and profit of $358,724.50045

Answer 2

I'll give you some hints to start but you need to address you mind to the mechanics of the problem. The application of course is to microeconomics.

Hints:
The quantity of a product demanded (sold) is a function of price.
Total revenue is the product of price and quantity sold
Total cost is equal to fixed costs plus variable costs.
Variable cost is a function of output (quantity produced), inter alia.
Revenue is maximized where marginal revenue equals zero (in this case)
Profit maximization occurs where marginal cost equals marginal revenue.
The first derivative of the total cost and total revenue functions will give you marginal cost and marginal revenue respectively.

You should have everything you need to crunch out your answers!

Answer 3

a)...C(q) = 2,000 + 3.50*x b)...R(q) = 10x c).. 10x = 2000 + 3.50x ......6.50x= 2000 = ..... x = 307.69 at this point revenue exactly eqwua ls manufacturing costs d) P(q) = R(q) - C(q) If 150 G-pods are sold substitute 150 for x in the equations above

Answer 4

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