how to find revenue function

Sample Problems, Lesson 4.7

Sample Problem: #10, Lesson 4.7

For the given cost and demand function, find the production level that will maximize profit.	Given Problem: #10, Lesson 4.7
	We know that to maximize profit, marginal revenue must equal marginal cost . This means we need to find C'(x) (marginal cost) and we need the Revenue function and its derivative, R'(x) (marginal revenue).
	To maximize profit, we need to set marginal revenue equal to the marginal cost, and solve for x. We find that when 100 units are produced, that profit is currently maximized.
	To check our work, we set up the Profit function first.
	Using the profit function in the last step a spreadsheet can be set up to calculate the profit at various production levels. We can see that this confirms what we found above. The chart shows that at 100 units (x), we have a maximum profit of 46000. Note that if x continues to increase, the profit declines.

Example Problem: #12, Lesson 4.7

Find the production level at which the marginal cost function starts to increase.

Sample Problem: #12, Lesson 4.7

We find not only the marginal cost C'(x), but also its derivative which would give us the rate of change of the marginal cost. We then find when the rate of change is zero. Following this, the marginal cost would have to increase.

We can see that the marginal cost function is an upright parabola. We could have also found the equation of the axis of symmetry, and again we would have found that the marginal cost begins to increase when we have produced 417 units.

Example Problem: #14, Lesson 4.7

An aircraft manufacturer wants to determine the best selling price for a new airplane. The company estimates that the initial cost of designing the airplane and setting up the factories in which to build it will be 500 million dollars. The additional cost of manufacturing each plane can be modeled by the function: , where x is the number of aircraft produced and m is the manufacturing cost, in millions of dollars. The company estimates that if it charges a price p (in millions of dollars) for each plane, it will be able to sell planes. (a) Find the cost, demand, and revenue functions. (b) Find th eproduction level and the associated selling price of the aircraft that maximizes profit.	Given Problem, #14, Lesson 4.7
	part (a) The cost function is simply the intial cost plus the manufacturing cost. The demand function was given to us. The revenue function is simply x multiplied by the demand function.
	part (b) We know that to maximize profit, marginal revenue must equal marginal cost . This means we need to find C'(x) (marginal cost) and we need the Revenue function and its derivative, R'(x) (marginal revenue).
	To maximize profit, we need to set marginal revenue equal to the marginal cost, and solve for x. (because of the 4th root, a computer algebra system is used to solve for x.) Here is the maple command which allowed us to get the approximate solution: > *solve(-80+4.112x=x^(-.25),x); We find that when 20 planes are produced, that profit is currently maximized.**