Given the total cost equation: TC = 144 +2Q+ Q² what is the average cost when the firm produces at the level of Q that yields the minimum average cost?
Given the total cost equation: TC = 144 +2Q+ Q² what is the average cost when the firm produces at the level of Q that yields the minimum average cost?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![Given the total cost equation: TC = 144 + 2Q + Q², what is the average cost when the firm produces at the level of Q that yields the minimum average cost?
To find the minimum average cost, follow these steps:
1. **Understand the Formulas:**
- Total Cost (TC) is given by the equation TC = 144 + 2Q + Q².
- Average Cost (AC) is the Total Cost divided by the quantity (Q), or AC = TC/Q.
2. **Calculate Average Cost:**
- Substitute the TC expression into AC:
\[
AC = \frac{144 + 2Q + Q^2}{Q} = \frac{144}{Q} + 2 + Q
\]
3. **Find Minimum Average Cost:**
- To find the quantity (Q) that minimizes the average cost, take the first derivative of AC with respect to Q and set it to zero:
\[
\frac{d(AC)}{dQ} = -\frac{144}{Q^2} + 1 = 0
\]
- Solve for Q:
\[
-\frac{144}{Q^2} + 1 = 0 \implies \frac{144}{Q^2} = 1 \implies Q^2 = 144 \implies Q = 12
\]
- Check the second derivative to ensure it is a minimum:
\[
\frac{d^2(AC)}{dQ^2} = \frac{288}{Q^3}
\]
This is positive for Q = 12, confirming a minimum.
4. **Substitute Q into the AC Formula:**
- AC = \(\frac{144}{12} + 2 + 12 = 12 + 2 + 12 = 26\)
Thus, the minimum average cost occurs when the firm produces 12 units, yielding an average cost of 26.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F463d895f-c496-4b53-b6f0-f602af7f16a5%2Fdbdbe7c4-4a5f-4ef5-9190-105e1646b697%2Fv74wq4dd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Given the total cost equation: TC = 144 + 2Q + Q², what is the average cost when the firm produces at the level of Q that yields the minimum average cost?
To find the minimum average cost, follow these steps:
1. **Understand the Formulas:**
- Total Cost (TC) is given by the equation TC = 144 + 2Q + Q².
- Average Cost (AC) is the Total Cost divided by the quantity (Q), or AC = TC/Q.
2. **Calculate Average Cost:**
- Substitute the TC expression into AC:
\[
AC = \frac{144 + 2Q + Q^2}{Q} = \frac{144}{Q} + 2 + Q
\]
3. **Find Minimum Average Cost:**
- To find the quantity (Q) that minimizes the average cost, take the first derivative of AC with respect to Q and set it to zero:
\[
\frac{d(AC)}{dQ} = -\frac{144}{Q^2} + 1 = 0
\]
- Solve for Q:
\[
-\frac{144}{Q^2} + 1 = 0 \implies \frac{144}{Q^2} = 1 \implies Q^2 = 144 \implies Q = 12
\]
- Check the second derivative to ensure it is a minimum:
\[
\frac{d^2(AC)}{dQ^2} = \frac{288}{Q^3}
\]
This is positive for Q = 12, confirming a minimum.
4. **Substitute Q into the AC Formula:**
- AC = \(\frac{144}{12} + 2 + 12 = 12 + 2 + 12 = 26\)
Thus, the minimum average cost occurs when the firm produces 12 units, yielding an average cost of 26.
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