ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN: 9780190931919
Author: NEWNAN
Publisher: Oxford University Press
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![### Understanding Supply and its Rate of Change with Respect to Price
To find the supply \( S(p) \) and the instantaneous rate of change of supply with respect to price when the price is 40 dollars, the following problem is presented:
\[ S(p) = \frac{100p}{0.7p + 3} \]
where \( p \) is the price in dollars.
1. **Calculate the Supply at \( p = 40 \) Dollars**
\[ S(40) = \_\_\_\_\_ \text{ players} \]
2. **Calculate the Instantaneous Rate of Change of Supply with Respect to Price at \( p = 40 \) Dollars**
\[ S'(40) = \_\_\_\_\_ \text{ players/dollar} \]
3. **Estimate the Increase in Supply if the Price Rises from \( p = 40 \) to \( p = 50 \) Dollars**
Use your answers from parts 1 and 2 to estimate the increase in supply. Round your answer to the nearest integer.
\[ \text{Supply increase} \approx \_\_\_\_\_ \text{ players} \]
This exercise helps in understanding how changes in price can impact supply, an important concept in economic theory.
### Explanation of the Procedure and Estimations
To solve the problem step by step:
- **Step 1: Calculate \( S(40) \)**
Substitute \( p = 40 \) into the supply function:
\[ S(40) = \frac{100 \times 40}{0.7 \times 40 + 3} = \frac{4000}{28 + 3} = \frac{4000}{31} \]
Solve to get the supply at 40 dollars.
- **Step 2: Calculate \( S'(40) \)**
Differentiate \( S(p) \) with respect to \( p \) to find \( S'(p) \). This involves applying the quotient rule of differentiation. Then, substitute \( p = 40 \) into \( S'(p) \).
- **Step 3: Estimate Increase in Supply**
Using the derivative \( S'(p) \) at \( p = 40 \), estimate the increase in supply when the price increases from 40 to 50 dollars with the following approximation:
\[ \text](https://content.bartleby.com/qna-images/question/1d39eb1c-3154-4bee-b19a-a1be6277f6e8/9fae7bfb-2158-4fd6-88de-f87469301aeb/oewndcgi_processed.png)
Transcribed Image Text:### Understanding Supply and its Rate of Change with Respect to Price
To find the supply \( S(p) \) and the instantaneous rate of change of supply with respect to price when the price is 40 dollars, the following problem is presented:
\[ S(p) = \frac{100p}{0.7p + 3} \]
where \( p \) is the price in dollars.
1. **Calculate the Supply at \( p = 40 \) Dollars**
\[ S(40) = \_\_\_\_\_ \text{ players} \]
2. **Calculate the Instantaneous Rate of Change of Supply with Respect to Price at \( p = 40 \) Dollars**
\[ S'(40) = \_\_\_\_\_ \text{ players/dollar} \]
3. **Estimate the Increase in Supply if the Price Rises from \( p = 40 \) to \( p = 50 \) Dollars**
Use your answers from parts 1 and 2 to estimate the increase in supply. Round your answer to the nearest integer.
\[ \text{Supply increase} \approx \_\_\_\_\_ \text{ players} \]
This exercise helps in understanding how changes in price can impact supply, an important concept in economic theory.
### Explanation of the Procedure and Estimations
To solve the problem step by step:
- **Step 1: Calculate \( S(40) \)**
Substitute \( p = 40 \) into the supply function:
\[ S(40) = \frac{100 \times 40}{0.7 \times 40 + 3} = \frac{4000}{28 + 3} = \frac{4000}{31} \]
Solve to get the supply at 40 dollars.
- **Step 2: Calculate \( S'(40) \)**
Differentiate \( S(p) \) with respect to \( p \) to find \( S'(p) \). This involves applying the quotient rule of differentiation. Then, substitute \( p = 40 \) into \( S'(p) \).
- **Step 3: Estimate Increase in Supply**
Using the derivative \( S'(p) \) at \( p = 40 \), estimate the increase in supply when the price increases from 40 to 50 dollars with the following approximation:
\[ \text
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