Daniel's Artworks faces a short-run total cost of production given by: - TC = Q³ – 12Q² + 100Q + 1000, where Q is number of artworks produced per day. Daniel's marginal cost of producing artworks is: 3Q² – 24Q + 100. If the artworks sell for $60 per artwork, how many artworks should Daniel produce?
Daniel's Artworks faces a short-run total cost of production given by: - TC = Q³ – 12Q² + 100Q + 1000, where Q is number of artworks produced per day. Daniel's marginal cost of producing artworks is: 3Q² – 24Q + 100. If the artworks sell for $60 per artwork, how many artworks should Daniel produce?
Economics (MindTap Course List)
13th Edition
ISBN:9781337617383
Author:Roger A. Arnold
Publisher:Roger A. Arnold
Chapter21: Production And Costs
Section21.4: Costs Of Production: Total, Average, Marginal
Problem 3ST
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![**Title: Cost Analysis for Daniel's Art Production**
Daniel's Artworks faces a short-run total cost of production given by:
\[ TC = Q^3 - 12Q^2 + 100Q + 1000 \]
where \( Q \) is the number of artworks produced per day.
Daniel’s marginal cost of producing artworks is:
\[ 3Q^2 - 24Q + 100 \]
If the artworks sell for $60 per artwork, how many artworks should Daniel produce?
**Explanation:**
- **Total Cost (TC) Function:** This equation represents the total cost concerning the output \( Q \). It is a cubic polynomial indicating how costs are incurred as production varies.
- **Marginal Cost (MC) Function:** This is the derivative of the total cost function. It indicates the cost of producing one additional artwork, highlighting changes in production cost with respect to output.
To determine how many artworks Daniel should produce, we set the marginal cost equal to the price per artwork ($60) and solve for \( Q \):
\[ 3Q^2 - 24Q + 100 = 60 \]
By solving this equation, Daniel finds the optimal production quantity where marginal cost equals price, maximizing profit.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9ea1563-4d06-4ae6-a752-621b00eac3e5%2Fa3990b1d-3f89-4984-ab8f-17fa73228aa4%2Fnpbg4nm_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Cost Analysis for Daniel's Art Production**
Daniel's Artworks faces a short-run total cost of production given by:
\[ TC = Q^3 - 12Q^2 + 100Q + 1000 \]
where \( Q \) is the number of artworks produced per day.
Daniel’s marginal cost of producing artworks is:
\[ 3Q^2 - 24Q + 100 \]
If the artworks sell for $60 per artwork, how many artworks should Daniel produce?
**Explanation:**
- **Total Cost (TC) Function:** This equation represents the total cost concerning the output \( Q \). It is a cubic polynomial indicating how costs are incurred as production varies.
- **Marginal Cost (MC) Function:** This is the derivative of the total cost function. It indicates the cost of producing one additional artwork, highlighting changes in production cost with respect to output.
To determine how many artworks Daniel should produce, we set the marginal cost equal to the price per artwork ($60) and solve for \( Q \):
\[ 3Q^2 - 24Q + 100 = 60 \]
By solving this equation, Daniel finds the optimal production quantity where marginal cost equals price, maximizing profit.
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