/*! 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 For the following exercises, wri... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 6

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as the square root of \(x\) and when \(x=36, y=24\).

Short Answer

Expert verified
The equation is \( y = 4 \sqrt{x} \).

Step by step solution

01

Understand direct variation

When we say that a variable \( y \) varies directly as the square root of \( x \), it means we can express their relationship using the equation \( y = k \sqrt{x} \), where \( k \) is a constant. Our task is to find the value of the constant \( k \).
02

Substitute known values

We are given that when \( x = 36 \), \( y = 24 \). Substitute these values into the equation \( y = k \sqrt{x} \) to determine \( k \). So, \( 24 = k \sqrt{36} \).
03

Calculate \( \sqrt{x} \)

Find the square root of \( x \). In this case, \( \sqrt{36} = 6 \).
04

Solve for \( k \)

Substitute \( \sqrt{36} = 6 \) into the equation from Step 2: \( 24 = k (6) \). Divide both sides by 6 to solve for \( k \). This gives \( k = \frac{24}{6} = 4 \).
05

Write the final equation

Now that we have \( k \), substitute it back into the equation \( y = k \sqrt{x} \). The relationship between \( y \) and \( x \) is therefore \( y = 4 \sqrt{x} \).

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.

Square Root
The concept of a square root is fundamental in algebra. When we talk about the square root of a number, we're referring to a value that, when multiplied by itself, gives the original number. For example, the square root of 36 is 6 because 6 multiplied by itself (6 脳 6) equals 36. This operation helps in simplifying equations and understanding relationships between variables.

  • Notion: The square root is represented by the radical symbol (鈭�).
  • Use in equations: Solves many algebraic problems by simplifying numbers under the radical.
  • Relation to direct variation: Shows how a variable can directly change as another's square root changes.
In direct variation problems, understanding square roots allows us to describe how one variable changes with respect to another. This is crucial when forming and solving equations.
Constant of Variation
The constant of variation, represented by the symbol \( k \), is a key component in direct variation equations. It serves as the proportionality constant that describes how much one variable changes with respect to a change in another.

  • Definition: \( k \) is the multiplier that scales the relationship between variables.
  • Role: Determines how steep or shallow this variation is.
  • Dependence: On the initial conditions or values given in the problem.
In the context of direct variation like \( y = k \sqrt{x} \), \( k \) tells us the rate at which \( y \) changes with respect to changes in the square root of \( x \). This serves to tailor the basic variation formula to accurately model the given data.
Algebraic Equation
An algebraic equation is a mathematical statement that shows the equality between two expressions. These equations are essential in solving for unknown values and understanding relationships between variables.

  • Structure: Composed of variables, constants, and operations such as addition, multiplication, and roots.
  • Purpose: To express relationships and solve for unknowns.
  • Example: In our exercise, the equation \( y = k \sqrt{x} \) shows the direct relationship between \( y \) and the square root of \( x \).
In practice, algebraic equations like the one above are used to model real-world situations. They allow students to translate word problems into mathematical statements that can be manipulated to find solutions.
Solving for k
To find the constant of variation \( k \) in an algebraic equation, like \( y = k \sqrt{x} \), the process involves substituting known values and algebraic manipulation.

  • Substitute given values into the equation to replace variables with numbers.
  • Calculate necessary components, like square roots, before solving for \( k \).
  • Isolate \( k \) by performing inverse operations.
In our problem where \( x = 36 \) and \( y = 24 \), you substitute these values into the equation to get \( 24 = k (6) \). Solving this equation by dividing both sides by 6 gives \( k = 4 \). This process isolates \( k \) and gives you the exact proportion necessary to maintain the direct variation described.

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

For the following exercises, find the dimensions of the right circular cylinder described. The radius and height differ by two meters. The height is greater and the volume is 28.125\(\pi\) cubic meters.

Find the dimensions of the box described. The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.

For the following exercises, use the given information to answer the questions. The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as \(x\) and \(z\) and inversely as \(w .\) When \(x=5, z=2,\) and \(w=20,\) then \(y=4 .\) Find \(y\) when \(x=3\) and \(z=8,\) and \(w=48\).

For the following exercises, determine the function described and then use it to answer the question. Consider a cone with height of 30 feet. Express the radius, \(r\) , in terms of the volume, \(V\) , and find the radius of a cone with volume of 1000 cubic feet.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.