Question 1A Clothing Manufacturer Has Determined That The Weekly Demand Function Of Their Sweaters Is Given By $P=f(q)=144-q^2$, Where $P$ Is Measured In Botswana Pula And $q$ Is Measured In Units Of Thousand.a. Find The

Introduction

In the world of business, revenue maximization is a crucial goal for companies to achieve. One way to achieve this is by understanding the demand function of a product and using it to determine the optimal price and quantity to sell. In this article, we will analyze the weekly demand function of a clothing manufacturer's sweaters, given by $P=f(q)=144-q^2$, where $P$ is measured in Botswana Pula and $q$ is measured in units of thousand. Our objective is to find the revenue function and determine the quantity that maximizes revenue.

The Demand Function

The demand function is a mathematical representation of the relationship between the price of a product and the quantity demanded. In this case, the demand function is given by $P=f(q)=144-q^2$. This function tells us that the price of a sweater is a function of the quantity demanded, and that the price decreases as the quantity demanded increases.

Revenue Function

The revenue function is a mathematical representation of the total revenue generated by selling a product. It is calculated by multiplying the price of the product by the quantity sold. In this case, the revenue function is given by $R(q)=P(q)q=f(q)q=(144-q^2)q=144q-q^3$.

Finding the Maximum Revenue

To find the maximum revenue, we need to find the critical points of the revenue function. Critical points occur when the derivative of the function is equal to zero or undefined. In this case, we need to find the derivative of the revenue function with respect to $q$.

Derivative of the Revenue Function

Using the power rule of differentiation, we can find the derivative of the revenue function as follows:

$$\frac{dR}{dq}=\frac{d}{dq}(144q-q^3)=144-3q^2$$

Setting the Derivative Equal to Zero

To find the critical points, we set the derivative equal to zero and solve for $q$:

$$144-3q^2=0$$

Solving for $q$, we get:

$$3q^2=144$$

$$q^2=48$$

$$q=\pm\sqrt{48}$$

Since the quantity demanded cannot be negative, we take the positive square root:

$$q=\sqrt{48}=4\sqrt{3}$$

Second Derivative Test

To determine whether this critical point corresponds to a maximum, minimum, or saddle point, we need to use the second derivative test. The second derivative of the revenue function is given by:

$$\frac{d^2R}{dq^2}=-6q$$

Evaluating the second derivative at the critical point, we get:

$$\frac{d^2R}{dq^2}\bigg|_{q=4\sqrt{3}}=-6(4\sqrt{3})=-24\sqrt{3}<0$$

Since the second derivative is negative, we conclude that the critical point corresponds to a maximum.

Conclusion

In this article, we analyzed the weekly demand function of a clothing manufacturer's sweaters, given by $P=f(q)=144-q^2$. We found the revenue function and determined the quantity that maximizes revenue. The critical point of the revenue function was found to be $q=4\sqrt{3}$, and the second derivative test confirmed that this corresponds to a maximum. This analysis can be used by the clothing manufacturer to determine the optimal price and quantity to sell in order to maximize revenue.

Recommendations

Based on this analysis, the clothing manufacturer should aim to sell $4\sqrt{3}$ thousand units of sweaters per week in order to maximize revenue. This can be achieved by setting the price of the sweaters at $P=f(q)=144-q^2$ and selling the optimal quantity.

Limitations

This analysis assumes that the demand function is given by $P=f(q)=144-q^2$ and that the revenue function is given by $R(q)=P(q)q=f(q)q=(144-q^2)q=144q-q^3$. In reality, the demand function and revenue function may be more complex and may depend on various factors such as seasonality, competition, and consumer preferences. Therefore, this analysis should be used as a starting point and should be modified and refined based on actual data and market conditions.

Future Research Directions

Introduction

In our previous article, we analyzed the weekly demand function of a clothing manufacturer's sweaters, given by $P=f(q)=144-q^2$. We found the revenue function and determined the quantity that maximizes revenue. In this article, we will answer some frequently asked questions related to revenue maximization in the sweater industry.

Q: What is the optimal price and quantity to sell in order to maximize revenue?

A: The optimal price and quantity to sell in order to maximize revenue are $P=f(q)=144-q^2$ and $q=4\sqrt{3}$ thousand units of sweaters per week, respectively.

Q: How can the clothing manufacturer determine the optimal price and quantity to sell?

A: The clothing manufacturer can determine the optimal price and quantity to sell by analyzing the demand function and revenue function. The demand function tells us how the price of the sweaters changes as the quantity demanded increases, while the revenue function tells us how the total revenue generated by selling the sweaters changes as the quantity sold increases.

Q: What are the limitations of this analysis?

A: This analysis assumes that the demand function is given by $P=f(q)=144-q^2$ and that the revenue function is given by $R(q)=P(q)q=f(q)q=(144-q^2)q=144q-q^3$. In reality, the demand function and revenue function may be more complex and may depend on various factors such as seasonality, competition, and consumer preferences.

Q: How can the clothing manufacturer extend this analysis to include other factors such as seasonality, competition, and consumer preferences?

A: The clothing manufacturer can extend this analysis by using statistical techniques such as regression analysis to estimate the demand function and revenue function based on actual data. The analysis can also be extended to include other factors such as seasonality, competition, and consumer preferences by using techniques such as time series analysis and econometric modeling.

Q: What are the benefits of revenue maximization in the sweater industry?

A: The benefits of revenue maximization in the sweater industry include increased profitability, improved competitiveness, and enhanced customer satisfaction. By maximizing revenue, the clothing manufacturer can increase its profitability, improve its competitiveness in the market, and enhance customer satisfaction by offering high-quality products at competitive prices.

Q: What are the challenges of revenue maximization in the sweater industry?

A: The challenges of revenue maximization in the sweater industry include managing inventory levels, controlling costs, and responding to changes in demand and market conditions. The clothing manufacturer must carefully manage its inventory levels to ensure that it has sufficient stock to meet demand, while controlling costs to maintain profitability. The manufacturer must also be able to respond quickly to changes in demand and market conditions to remain competitive.

Q: How can the clothing manufacturer respond to changes in demand and market conditions?

A: The clothing manufacturer can respond to changes in demand and market conditions by using techniques such as market research, customer feedback, and data analysis to stay informed about changes in the market. The manufacturer can also use techniques such as flexible pricing and inventory management to respond quickly to changes in demand and market conditions.

Conclusion

In this article, we have answered some frequently asked questions related to revenue maximization in the sweater industry. We have discussed the optimal price and quantity to sell in order to maximize revenue, the limitations of this analysis, and the benefits and challenges of revenue maximization in the sweater industry. We have also discussed how the clothing manufacturer can extend this analysis to include other factors such as seasonality, competition, and consumer preferences, and how the manufacturer can respond to changes in demand and market conditions.