# Finding Profit-Maximizing Level of Production

1. Groovy Tuesday, a clothing manufacturer, has found that their costs can be approximated by the equation: C = 500 + 2Q2. The consulting firm they hired to estimate their current demand determined that demand is characterized by: Q = 450 - 2P. What level of production would maximize their profit?

2. Musical Melodies, a sheet music retailer, recently hired a management consulting firm to identify ways in which they can increase profit. You, the head consultant on the Musical Melodies case, have begun this analysis by collecting data on quantity demanded, average price of Musical Melodies music, the average price of a sheet of music through Crystal Copies, the average income of consumers, and advertising expenditures. You have estimated that their demand is characterized by the equation Q = 6,000 - 3P. The management team has provided you with the cost equation of C = 650 + 4.5Q. Find the profit-maximizing level of production.

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#### Solution Preview

1. The firm's profit will be maximized when MR equals MC, which is the profit maximizing condition.

We find marginal cost function by differentiating the cost function:

C = 500 +2Q^2

Then MC = 4Q

Now find the inverse demand function:

P = 450/2 - Q/2 = 225 -Q/2

Multiply ...

#### Solution Summary

Finding the profit-maximizing level of production given cost and demand functions.