Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problema 64

Capitolo 4, Problema 64

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

Guida verificata passo dopo passo

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

Express one variable in terms of the other using the difference equation. For example,

x = y + 24

Write the product of the two numbers as a function of one variable:

P(y) = x � y = (y + 24) 	imes y = y^2 + 24 y .

To find the minimum product, consider

P(y) = y^2 + 24 y

as a quadratic function and find its vertex. Recall that the vertex of

ax^2 + bx + c

is at

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

. Here, identify

a

and

b

and calculate the value of

y

at the vertex.

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

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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.

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Introduction to Rational Equations

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.

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Properties of Parabolas

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.

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Solving Quadratic Equations Using The Quadratic Formula

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