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.

Total cost is sum total of the fixed cost and variable cost. The average cost is calculated by the…

Answered: Given the total cost equation: TC = 144…

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?

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

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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.

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