## Managerial Calc: Derivatives & Cost Functions

I've been assigned this optional bonus problem that I cannot quite figure out entirely and goes as follows:

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The Apex company has determined that for one month the marginal cost function for its round widgets is C'(x) = 0.5x + 50 where x represents the number of boxes of widgets which can be produced each month and C is in dollars. Each box contains 120 widgets. The cost of producing 200 boxes of widgets is $22,500. Also, if they charge$50 per box, they will sell 350 boxes; and if they charge \$75 per box, they will sell 300 boxes. Assume that the demand equation is linear.

1.

Find:
(a) the demand equation
(b) the revenue function
(c) the number of boxes which they should sell in order to maximize the revenue
(d) the maximum revenue
(e) the price which should be charged in order to maximize the revenue
(f) the (total) cost function
(g) the average cost function
(h) the production level (number of boxes) which yields the minimum average cost
(i) the minimum average cost
(j) the profit function
(k) the number of boxes which they should produce and sell in order to maximize profit
(l) the maximum profit
(m) the price which they should be charged to maximize profits

2.

Suppose that the plant capacity is such that not more than 200 boxes of round widgets can be produced each month. Which one of your answers to question 1 will be changed? Explain how. Find what the new answer to these questions will be.

3. Suppose that the boxes of round widgets cannot be opened and divided before being shipped to the customer. Which of your questions to question 1 will be changed? Explain how. Find what the new answers to these questions will be.

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I will be very appreciative of any help I can get on this. So far I've managed to get a few thing which are:

The cost function
c(x) = .25^2 + 50x + 2500

The profit function:
[ -(1/2x^2) + 225x ] - ( .25x^2 + 50x + 2500)
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