/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Write an equation that expresses... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 6

Write an equation that expresses the statement. \(P\) varies inversely as \(T\)

Short Answer

Expert verified
The equation is \(P = \frac{k}{T}\).

Step by step solution

01

Understand the Concept of Inverse Variation

When a variable varies inversely as another variable, it means that as one variable increases, the other decreases, such that their product is constant. Mathematically, this is expressed as \(P \times T = k\), where \(k\) is a constant.
02

Express the Inverse Variation Equation

Based on the definition of inverse variation, write the equation that represents this relationship. The equation is \(P \times T = k\), where \(k\) is a constant. If you solve for \(P\), the equation becomes \(P = \frac{k}{T}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Direct Variation
Direct Variation is a concept in mathematics where two variables change in the same direction. This means that when one variable increases, the other also increases, and when one decreases, the other does too. The relationship is linear, and it can be written as \( y = kx \), where \( y \) and \( x \) are the variables and \( k \) is a constant called the constant of variation or proportionality.
If you imagine driving a car, a direct variation scenario might occur with speed and distance. The faster you drive, the further you go in the same amount of time.
  • Key Points About Direct Variation: The graph of a direct variation passes through the origin (0,0) because the relationship starts from zero.
  • The constant \( k \) can be found by dividing \( y \) by \( x \) when given a set of values.
  • If two quantities vary directly, the ratio of one to the other is always the same.
Understanding direct variation helps differentiate from inverse variation, where variables move in opposite directions.
Mathematical Equations
Mathematical equations are statements that show the equality of two expressions. They are the foundation of many precalculus concepts as they allow us to represent real-world situations in a structured form.
When solving equations, we are finding values that make the equation true. In the case of inverse variation, the equation \( P \times T = k \) is used to express that the product of \( P \) and \( T \) remains constant.
  • Components of an Equation: You usually have variables (like \( P \) and \( T \)), numbers, and operation signs.
  • An equation can be interpreted by isolating terms to find unknown values.
  • Equations help to illustrate relationships between quantities, making it easier to predict one when the other changes.
Equations are powerful tools in math, enabling us to solve problems in various fields such as physics, engineering, and economics.
Precalculus Concepts
Precalculus is a branch of mathematics that prepares students for calculus. It includes the study of functions, complex numbers, and basic mathematical principles necessary for calculus.
Inverse and direct variations are part of these principles, helping to build a foundational understanding of how variables can interact with one another.
  • The Purpose of Precalculus: It provides necessary skills for understanding limits, derivatives, and integrals in calculus.
  • It covers various functions and models that describe real-world phenomena.
  • Prepares students to think analytically about how changes in values affect other related values.
By grasping concepts such as variation, students gain insight into more complex mathematical theories and real-world applications, paving the way for success in calculus studies.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Factor the expression completely. (This type of expression arises in calculus when using the "product rule.") $$5\left(x^{2}+4\right)^{4}(2 x)(x-2)^{4}+\left(x^{2}+4\right)^{5}(4)(x-2)^{3}$$

Factor the expression completely. $$9 x^{2}-36 x-45$$

Use a Special Factoring Formula to factor the expression. $$16 z^{2}-24 z+9$$

Use a Special Factoring Formula to factor the expression. $$8 s^{3}-125 t^{6}$$

Factor the expression completely. $$\left(1+\frac{1}{x}\right)^{2}-\left(1-\frac{1}{x}\right)^{2}$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.