Among all pairs of numbers whose difference is 24, find a pair whose product is as small as possible. What is the minimum product?

Verified step by step guidance

1. Let the two numbers be $x$ and $y$. According to the problem, their difference is 24, so we can write the equation $x-y=\; 24$.
2. Express one variable in terms of the other using the difference equation. For example, $x=y+\; 24$.
3. Write the product of the two numbers as a function of one variable: $P(y)\; =x�y=\; (y+\; 24)\; imesy=y^2\; +\; 24y.$
4. To find the minimum product, consider $P(y)\; =y^2\; +\; 24y$ as a quadratic function and find its vertex. Recall that the vertex of $ax^2\; +\; bx\; +\; c$ is at $x\; =\; -\backslash (\backslash frac\{b\}\{2a\}\backslash )$. Here, identify $a$ and $b$ and calculate the value of $y$ at the vertex.
5. Substitute the value of $y$ found in the previous step back into the product function $P(y)$ to find the minimum product. Then, use $x=y+\; 24$ to find the corresponding $x$.

Key Concepts

Formulating Equations from Word Problems

This involves translating the given conditions into algebraic expressions or equations. For example, if two numbers differ by 24, we can represent one number as x and the other as x + 24. This step is crucial to set up the problem for further analysis.

Quadratic Functions and Their Properties

The product of two numbers expressed in terms of one variable often forms a quadratic function. Understanding the shape of a parabola and how to find its minimum or maximum value using vertex formulas or completing the square is essential to solve optimization problems.

Optimization Using Derivatives or Vertex Formula

To find the minimum or maximum value of a quadratic function, one can use calculus (derivatives) or algebraic methods (vertex formula). The vertex of the parabola gives the point where the product is minimized or maximized, which directly answers the problem.