### Cost Minimization in Manufacturing**Problem Statement:**Assume that it costs a company approximately\[ C(x) = 484,000 + 160x + 0.001x^2 \]dollars to manufacture $ x $ units of a device in an hour at one of their manufacturing centers. 1. **How many devices should be manufactured each hour to minimize average cost?** - **Answer:** \[ \_\_\_ \text{ units} \]2. **What is the resulting average cost of a device?** - **Answer:** \[ \$\_\_\_ \]3. **How does the average cost compare with the marginal cost at the optimal production level? Find how much they differ.** - **Answer:** \[ \$\_\_\_ \]**Explanation:**To find the optimal number of devices to minimize average cost, you must first derive the function for the average cost $ AC(x) $ and then determine the number of units that minimize this cost. Additionally, comparing the marginal cost $ MC(x) $ at the optimal production level against the average cost provides insight into the efficiency of the production.- Average Cost Function $ AC(x) $: \[ AC(x) = \frac{C(x)}{x} = \frac{484,000 + 160x + 0.001x^2}{x} = 484,000x^{-1} + 160 + 0.001x \]- Marginal Cost Function $ MC(x) $: The derivative of the total cost function $ C(x) $ gives the marginal cost: \[ MC(x) = \frac{dC(x)}{dx} = 160 + 0.002x \]To find the point where the average cost is minimized, set the derivative of $ AC(x) $ (Average Cost) equal to zero and solve for $ x $. Finally, evaluate both $ AC(x) $ and $ MC(x) $ at this optimal production level and compare the two to find the difference.

Marginal Cost: it refers to the change in total cost due to 1 additional unit of production. As due…

Assume that it costs a company approximately C(x) = 484,000+ 160x + 0.001x² dollars to manufacture x units of a device in an hour at one of their manufacturing centers. How many devices should be manufactured each hour to minimize average cost? units What is the resulting average cost of a device? $ How does the average cost compare with the marginal cost at the optimal production level? Find how much they differ. $

Assume that it costs a company approximately C(x) = 484,000+ 160x + 0.001x² dollars to manufacture x units of a device in an hour at one of their manufacturing centers. How many devices should be manufactured each hour to minimize average cost? units What is the resulting average cost of a device? $ How does the average cost compare with the marginal cost at the optimal production level? Find how much they differ. $

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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