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    Problem 1
    In deciding whether to set up a new manufacturing plant, company analysts have decided that a linear function is a reasonable estimation for the total cost C(x) to produce x items. They estimate the cost to produce 10,000 items as $547,500 and the cost to produce 50,000 items as $737,500
    (a) Find the formula for C(x)
    (b) Find the total cost to produce 100,000 items
    (c) Find the total cost to produce 0 items
    (d) Find the marginal cost of the items to be produced in this plant.
    You are the manager of a firm. You are considering the manufacture of a new product, so you ask the accounting department for cost estimates and the sales department for sales estimate. After you receive the data, you must decide whether to go ahead with production of the new product.
    Note: C(x) denotes the cost function and R(x) denotes the revenue function
    For problem 2 and 3 below do the following:
    (a) find a break even quantity and then
    (b) decide what you would do in each case (go ahead with the production or not ?).
    (c) Also write the profit function and identify the value of x, number of units, for which the profit is maximum.

    Problem 2
    C(x) = 105 x + 6000 ; R(x) = 250 x ; no more than 400 units can be sold.

    Problem 3
    C(x) = 1000x + 5000 ; R(x) = 900 x ; no restriction on the number of units that can be sold.
    In problem 4 and problem 5, let C(x) be the cost to produce x widgets and let R(x) be the revenue.
    For each problem 4 and 5 do the following:
    (a) find the break even quantity
    (b) find the maximum revenue
    (c) find the number of widgets, x, that generates the maximum revenue
    (d) find the maximum profit
    (e) find the number of widgets, x, that generates the maximum profit

    Problem 4
    R(x) = - + 8x
    C(x) = x + 15

    Problem 5
    R(x) = - 4 + 40x
    C(x) = 4x + 77

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    Solution Summary

    There are a variety of problems regarding solving systems of equations, including several optimization/total cost problems.