Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?

Verified step by step guidance

1. Let the two numbers be $x$ and $y$. According to the problem, their sum is 16, so we write the equation: $x+y=16$.
2. Express one variable in terms of the other using the sum equation. For example, solve for $y$: $y=16-x$.
3. Write the product $P$ of the two numbers as a function of $x$: $P(x)\; =ximesy=ximes\; (16\; -x)\; =\; 16x-x^2$.
4. To find the maximum product, recognize that $P(x)\; =\; -x^2\; +\; 16x$ is a quadratic function opening downward. Find the vertex of this parabola, which gives the maximum value. Use the vertex formula for $x$: $x\; =\; -\backslash frac\{b\}\{2a\}$, where $a\; =\; -1$ and $b\; =\; 16$.
5. After finding the value of $x$ at the vertex, substitute it back into $y\; =\; 16\; -\; x$ to find $y$. Then, calculate the product $P\; =\; x\; imes\; y$ to find the maximum product.

Key Concepts

Formulating the Problem Using Variables

To solve optimization problems, start by defining variables to represent the quantities involved. Here, if two numbers sum to 16, let one number be x and the other 16 - x. Expressing the product in terms of a single variable simplifies analysis.

Quadratic Functions and Their Properties

The product of the two numbers forms a quadratic function in terms of x. Understanding the shape of a parabola, which opens downward if the leading coefficient is negative, helps identify the maximum value by locating the vertex.

Finding the Vertex of a Parabola

The vertex of a quadratic function ax² + bx + c gives the maximum or minimum value. For a downward-opening parabola, the vertex represents the maximum. The x-coordinate of the vertex is found using -b/(2a), which helps determine the numbers yielding the maximum product.