Optimal Number of Purchase Orders

Oct 13, 2025 by thesweetdesigns.com 43 views

Understanding the Problem

The given function, $$f(x)=-(x-20)(x-100)$$, represents a company's monthly profit as a function of $$x$$, the number of purchase orders received. To find the number of purchase orders that will generate the greatest profit, we need to analyze the function and determine its maximum value.

Analyzing the Function

The function $$f(x)=-(x-20)(x-100)$$ is a quadratic function, which can be written in the form $$f(x)=ax^2+bx+c$$. In this case, the function is in the form of a product of two binomials, $$(x-20)$$ and $$(x-100)$$. To find the maximum value of the function, we need to find the vertex of the parabola represented by the function.

Finding the Vertex

The vertex of a parabola represented by the function $$f(x)=ax^2+bx+c$$ can be found using the formula $$x=-\frac{b}{2a}$$. In this case, the function is $$f(x)=-(x-20)(x-100)$$, which can be rewritten as $$f(x)=-x^2+120x-2000$$. Comparing this with the standard form of a quadratic function, we can see that $$a=-1$$ and $$b=120$$. Therefore, the x-coordinate of the vertex is $$x=-\frac{b}{2a}=-\frac{120}{2(-1)}=60$$.

Finding the Maximum Value

Now that we have found the x-coordinate of the vertex, we can find the maximum value of the function by substituting this value into the function. Substituting $$x=60$$ into the function $$f(x)=-(x-20)(x-100)$$, we get $$f(60)=-(60-20)(60-100)=-(40)(-40)=1600$$. Therefore, the maximum value of the function is $$1600$$, which occurs when $$x=60$$.

Conclusion

In conclusion, the number of purchase orders that will generate the greatest profit is $$60$$. This is because the function $$f(x)=-(x-20)(x-100)$$ represents a company's monthly profit as a function of $$x$$, the number of purchase orders received, and the maximum value of the function occurs when $$x=60$$.

Optimal Number of Purchase Orders

The optimal number of purchase orders is the number of purchase orders that will generate the greatest profit. In this case, the optimal number of purchase orders is $$60$$.

Comparison with Other Options

Let's compare the optimal number of purchase orders with the other options.

Option A: 20. This is the lower bound of the function, and the profit is $$0$$ when $$x=20$$.
Option B: 60. This is the optimal number of purchase orders, and the profit is $$1600$$ when $$x=60$$.
Option C: 80. This is greater than the optimal number of purchase orders, and the profit is less than the maximum value when $$x=80$$.
Option D: 100. This is the upper bound of the function, and the profit is $$0$$ when $$x=100$$.

Conclusion

In conclusion, the optimal number of purchase orders is $$60$$, and this is the number of purchase orders that will generate the greatest profit.

Real-World Applications

The concept of finding the optimal number of purchase orders can be applied to real-world scenarios. For example, a company may want to determine the optimal number of products to produce in order to maximize profit. By analyzing the function that represents the company's profit as a function of the number of products produced, the company can determine the optimal number of products to produce.

Limitations

One limitation of this problem is that it assumes that the function is a quadratic function. In reality, the function may be more complex, and the optimal number of purchase orders may not be a simple number. Additionally, the problem assumes that the company's profit is a function of the number of purchase orders received, but in reality, the company's profit may be influenced by other factors such as the price of the products and the cost of production.

Future Research

Future research could involve analyzing more complex functions that represent a company's profit as a function of the number of purchase orders received. Additionally, research could involve investigating the impact of other factors on a company's profit, such as the price of the products and the cost of production.

Conclusion

In conclusion, the optimal number of purchase orders is $$60$$, and this is the number of purchase orders that will generate the greatest profit. This problem can be applied to real-world scenarios, and it has limitations and potential for future research.

Optimal Number of Purchase Orders: A Summary

Option	Number of Purchase Orders	Profit
A	20	0
B	60	1600
C	80	Less than 1600
D	100	0

Conclusion

In conclusion, the optimal number of purchase orders is $$60$$, and this is the number of purchase orders that will generate the greatest profit.

Understanding the Problem

Q&A

Q: What is the function that represents a company's monthly profit?

A: The function is $$f(x)=-(x-20)(x-100)$$.

Q: What is the x-coordinate of the vertex of the parabola represented by the function?

A: The x-coordinate of the vertex is $$x=-\frac{b}{2a}=-\frac{120}{2(-1)}=60$$.

Q: What is the maximum value of the function?

A: The maximum value of the function is $$1600$$, which occurs when $$x=60$$.

Q: What is the optimal number of purchase orders?

A: The optimal number of purchase orders is $$60$$.

Q: Why is the optimal number of purchase orders 60?

A: The optimal number of purchase orders is 60 because the function $$f(x)=-(x-20)(x-100)$$ represents a company's monthly profit as a function of $$x$$, the number of purchase orders received, and the maximum value of the function occurs when $$x=60$$.

Q: What are the limitations of this problem?

A: One limitation of this problem is that it assumes that the function is a quadratic function. In reality, the function may be more complex, and the optimal number of purchase orders may not be a simple number. Additionally, the problem assumes that the company's profit is a function of the number of purchase orders received, but in reality, the company's profit may be influenced by other factors such as the price of the products and the cost of production.

Q: What are some real-world applications of this problem?

A: The concept of finding the optimal number of purchase orders can be applied to real-world scenarios. For example, a company may want to determine the optimal number of products to produce in order to maximize profit. By analyzing the function that represents the company's profit as a function of the number of products produced, the company can determine the optimal number of products to produce.

Q: What are some potential areas for future research?

A: Future research could involve analyzing more complex functions that represent a company's profit as a function of the number of purchase orders received. Additionally, research could involve investigating the impact of other factors on a company's profit, such as the price of the products and the cost of production.

Conclusion

Optimal Number of Purchase Orders: A Summary

Option	Number of Purchase Orders	Profit
A	20	0
B	60	1600
C	80	Less than 1600
D	100	0

Conclusion

In conclusion, the optimal number of purchase orders is $$60$$, and this is the number of purchase orders that will generate the greatest profit.