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Question: 1 a manufacturing company orders steel beams from a wholesaler...

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1. A manufacturing company orders steel beams from a wholesaler. Under the agreement with the wholesaler, the company must purchase at least 500 beams but no more than 2000, at a cost of $800 per beam. The company finds that when it purchases between 500 and 1000 beams, earns $1200 in revenue per beam. However, its per-beam revenue is reduced by 20 cents for each beam purchased in excess of 1000 beams (so for example if the company purchases 1011 beams it receives a revenue of $1197.80 for each of these 1011 beams) (a) Define the profit function P(x) as revenue minus costs (measured in dollars), where r is the number of beams purchased. Show that 00 1000 1000 < rS 2000. P(x) r) = ¡600r-0.2r2 (b) Is the function P( continuous for 5002000? Justify your answer. (c) Are there any values of a with 5002000 for which the derivative P() is not defined? If so, state what these values are. (d) Find the number of beams this company must purchase in order to maximise its profits. What is the maximum profit?
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