Overview
Problem 1: Cost and Marginal Analysis
A company’s total cost (in dollars) for producing units of a product is given by:
C(x) = 0.03x^3 - 4x^2 + 250x + 5000
1. Find the marginal cost function, .
2. Determine the production level that minimizes the marginal cost.
3. Calculate the marginal cost at this optimal production level.
4. Interpret the result and explain how it can help in cost management.
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Problem 2: Revenue and Profit Maximization
A company’s revenue function (in dollars) from selling units of a product is given by:
R(x) = 200x - 0.5x^2
The cost function is:
C(x) = 50x + 1000
1. Find the profit function , where .
2. Determine the critical points of and classify them as maxima or minima.
3. Find the number of units that should be produced and sold to maximize profit.
4. Calculate the maximum profit and explain its significance for the business.
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Problem 3: Optimization in Manufacturing
A manufacturer designs cylindrical cans to hold a fixed volume of 1000 cm³. The surface area (which determines material cost) is given by:
A(r) = 2\pi r h + 2\pi r^2
where is the radius and is the height of the can. The height is related to the volume by:
h = \frac{1000}{\pi r^2}
1. Express the surface area as a function of radius alone using the volume constraint.
2. Find the derivative and solve for the critical points to determine the optimal radius.
3. Determine the height corresponding to this optimal radius.
4. Explain why this radius and height combination minimizes material usage for the can.