Given cost and price (demand) functions C(q) = 110q+41,500 and p(q) = -1.9q+870, what is the marginal cost at a production level of 40 items? The marginal cost is dollars per item. (Round answer to nearest whole number.)
Given cost and price (demand) functions C(q) = 110q+41,500 and p(q) = -1.9q+870, what is the marginal cost at a production level of 40 items? The marginal cost is dollars per item. (Round answer to nearest whole number.)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Calculating Marginal Cost from Given Cost and Price Functions
**Problem Statement:**
Given cost and price (demand) functions \(C(q) = 110q + 41,500\) and \(p(q) = -1.9q + 870\), what is the marginal cost at a production level of 40 items?
**Solution:**
The marginal cost is \(\boxed{\phantom{9}} \) dollars per item.
(Round answer to nearest whole number.)
### Explanation:
The marginal cost is the additional cost incurred from producing one more unit of a good or service. It is derived from the cost function \(C(q)\) as follows:
1. **Differentiate the cost function \(C(q)\)** with respect to \(q\):
\[
\frac{d}{dq} (C(q)) = \frac{d}{dq} (110q + 41,500)
\]
Since the function \(C(q) = 110q + 41,500\) is linear:
\[
C'(q) = 110
\]
2. **Interpret the result**:
- The derivative, \(C'(q)\), represents the marginal cost.
- For \(C(q) = 110q + 41,500\), the marginal cost is constant and equal to \(110\) dollars per item, regardless of the production level.
**Conclusion:**
The marginal cost at any production level, including at a production level of 40 items, is \(110\) dollars per item.
Feel free to explore further topics such as changes in marginal cost with non-linear cost functions, or real-world applications of these calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12d36c83-f991-4c22-88cf-2adf2ca3b459%2F8d7cf2c5-1780-4728-9df9-e9dfbdeb0746%2Fk7bkf4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating Marginal Cost from Given Cost and Price Functions
**Problem Statement:**
Given cost and price (demand) functions \(C(q) = 110q + 41,500\) and \(p(q) = -1.9q + 870\), what is the marginal cost at a production level of 40 items?
**Solution:**
The marginal cost is \(\boxed{\phantom{9}} \) dollars per item.
(Round answer to nearest whole number.)
### Explanation:
The marginal cost is the additional cost incurred from producing one more unit of a good or service. It is derived from the cost function \(C(q)\) as follows:
1. **Differentiate the cost function \(C(q)\)** with respect to \(q\):
\[
\frac{d}{dq} (C(q)) = \frac{d}{dq} (110q + 41,500)
\]
Since the function \(C(q) = 110q + 41,500\) is linear:
\[
C'(q) = 110
\]
2. **Interpret the result**:
- The derivative, \(C'(q)\), represents the marginal cost.
- For \(C(q) = 110q + 41,500\), the marginal cost is constant and equal to \(110\) dollars per item, regardless of the production level.
**Conclusion:**
The marginal cost at any production level, including at a production level of 40 items, is \(110\) dollars per item.
Feel free to explore further topics such as changes in marginal cost with non-linear cost functions, or real-world applications of these calculations.
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