Question ### Marginal Profit Analysis of a CompanyThe following graph represents the **marginal profit $ P'(x) $** for a company based on the number of products, $ x $, it produces. #### Graph Explanation- **X-axis**: Number of items produced - **Y-axis**: Marginal profit in dollars per item The graph can be described as follows:1. From 0 to 10 items, the marginal profit increases linearly from approximately 0 to 4 dollars per item.2. Between 10 and 30 items, the marginal profit remains constant at 4 dollars per item.3. From 30 to 40 items, the marginal profit decreases linearly from 4 dollars per item to 0 dollars per item.4. Between 40 and 60 items, the marginal profit remains constant at 0 dollars per item.5. From 60 to 70 items, the marginal profit decreases linearly from 0 dollars per item to -3 dollars per item.6. Beyond 70 items, the marginal profit increases again from -3 dollars per item to 0 dollars per item at 80 items.#### Questions**a. How many products should the company sell to maximize its profit, $P(x)$?**To maximize the profit $ P(x) $, the company should produce and sell the number of items where the marginal profit $ P'(x) $ is highest and then becomes zero just before it turns negative. By analyzing the graph, the maximum profit is achieved at 40 items, where marginal profit slows down but is still positive.**b. Using your answer to part (a), find the maximum profit they will make.**To find the total maximum profit $ P(x) $, we integrate the marginal profit $ P'(x) $ from 0 to the point where the profit is maximized (from 0 to 40). Mathematically: \[ P(40) = \int_0^{40} P'(x) \, dx \]1. From 0 to 10: $\int_0^{10} (0.4x) \, dx = 0.4 \left[\frac{x^2}{2}\right]_0^{10} = 0.4 \left[ \frac{100}{2} - 0 \right] = 20 $2. From

Answered: 3. The graph below gives the marginal…

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3. The graph below gives the marginal profit P'(x) for a company when it produces x products. P'(x) ars per item +1

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

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Question

### Marginal Profit Analysis of a Company

The following graph represents the **marginal profit $ P'(x) $** for a company based on the number of products, $ x $, it produces.

#### Graph Explanation
- **X-axis**: Number of items produced
- **Y-axis**: Marginal profit in dollars per item

The graph can be described as follows:
1. From 0 to 10 items, the marginal profit increases linearly from approximately 0 to 4 dollars per item.
2. Between 10 and 30 items, the marginal profit remains constant at 4 dollars per item.
3. From 30 to 40 items, the marginal profit decreases linearly from 4 dollars per item to 0 dollars per item.
4. Between 40 and 60 items, the marginal profit remains constant at 0 dollars per item.
5. From 60 to 70 items, the marginal profit decreases linearly from 0 dollars per item to -3 dollars per item.
6. Beyond 70 items, the marginal profit increases again from -3 dollars per item to 0 dollars per item at 80 items.

#### Questions

**a. How many products should the company sell to maximize its profit, $P(x)$?**

To maximize the profit $ P(x) $, the company should produce and sell the number of items where the marginal profit $ P'(x) $ is highest and then becomes zero just before it turns negative. By analyzing the graph, the maximum profit is achieved at 40 items, where marginal profit slows down but is still positive.

**b. Using your answer to part (a), find the maximum profit they will make.**

To find the total maximum profit $ P(x) $, we integrate the marginal profit $ P'(x) $ from 0 to the point where the profit is maximized (from 0 to 40).

Mathematically:
\[ P(40) = \int_0^{40} P'(x) \, dx \]

1. From 0 to 10: $\int_0^{10} (0.4x) \, dx = 0.4 \left[\frac{x^2}{2}\right]_0^{10} = 0.4 \left[ \frac{100}{2} - 0 \right] = 20 $
2. From

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