Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problema 61

Capitolo 4, Problema 61

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

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Let the two numbers be

x

and

y

. According to the problem, their sum is 16, so we write the equation:

x + y = 16

Express one variable in terms of the other using the sum equation. For example, solve for

y

y = 16 - x

Write the product

P

of the two numbers as a function of

x

P(x) = x imes y = x imes (16 - x) = 16 x - x^2

To find the maximum product, recognize that

P(x) = - x^2 + 16 x

is a quadratic function opening downward. Find the vertex of this parabola, which gives the maximum value. Use the vertex formula for

x

x = -\(\frac{b}{2a}\)

, where

a = -1

and

b = 16

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.

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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.

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Equations with Two Variables

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.

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Properties of Parabolas

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.

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