Finding Two Numbers with a Specific Difference and Minimum Product
Mathematics often involves solving problems that require optimization, such as finding the minimum or maximum values of certain expressions. In this article, we will explore a problem where we need to find two numbers whose difference is -30 and whose product is minimum. This is a classic example that requires both algebraic manipulation and calculus to solve. We will delve into the step-by-step solution, use optimization techniques, and discuss the significance of the result.
Problem Statement
The problem statement is as follows: find two numbers such that the difference between them is -30 and their product is at a minimum.
Mathematical Formulation
To solve this problem, let's denote one of the numbers as x, and the other number as x - 30. This representation comes from the fact that the difference between the two numbers is given as -30. The product of these two numbers can be written as:
Product x × (x - 30)
Optimization Using Calculus
The next step involves using calculus to find the minimum value of the product. We start by expressing the product as a function of x:
Product x(x - 30) x^2 - 3
To find the critical points of this function, we need to take the derivative and set it to zero:
Derivative of Product d(Product)/dx 2x - 30
Setting the derivative equal to zero, we get:
2x - 30 0
Solving for x, we find:
x 15
This gives us one of the numbers as 15. The other number, which is 30 units less, is:
x - 30 15 - 30 -15
Conclusion and Product Calculation
Thus, the two numbers that satisfy the given conditions are 15 and -15. The product of these numbers is:
15 × (-15) -225
This value represents the minimum product.
Significance and Applications
The problem of finding the minimum product of two numbers with a fixed difference has practical applications in various fields, including economics, physics, and engineering. It can be used to solve real-world problems where minimizing a product is essential. For example, in economics, this might relate to cost minimization or maximizing profit margins.
By understanding these concepts, one can apply similar optimization techniques to solve more complex problems in fields like operations research, data analysis, and decision-making processes.