/*! 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 70 Solve for the indicated variable... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 70

Solve for the indicated variable. $$r=\sqrt[3]{\frac{3 V}{4 \pi}} \text { for } V$$

Short Answer

Expert verified
To solve the equation \(r = \sqrt[3]{\frac{3V}{4\pi}}\) for V, cube both sides of the equation to eliminate the cube root, giving \(r^3 = \frac{3V}{4\pi}\). Then, multiply both sides by 4蟺 and divide by 3 to isolate V. The final formula for V in terms of r is given by \(V = \frac{4}{3}\pi r^3\).

Step by step solution

01

Write the given formula

We have the formula: \(r = \sqrt[3]{\frac{3V}{4\pi}}\)
02

Cube both sides of the equation

To eliminate the cube root, we will cube both sides of the equation: \((r)^3 = \left(\sqrt[3]{\frac{3V}{4\pi}}\right)^3\)
03

Simplify the equation

The cube root and the cube cancel each other out, simplifying the equation to: \(r^3 = \frac{3V}{4\pi}\)
04

Solve for V

To isolate the variable V, we will multiply both sides of the equation by 4蟺: \(4\pi r^3 = 3V\) Now, divide both sides of the equation by 3 to solve for V: \(V = \frac{4}{3}\pi r^3\)
05

Write the final formula for V

Now that we have solved for V, we can write the final formula for the volume of a sphere in terms of its radius: \(V = \frac{4}{3}\pi r^3\) This formula gives the volume of a sphere with a given radius, r.

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.

Solving Equations
The process of solving equations involves finding the value(s) of the unknown variable(s) that make the equation true. In algebra, solving equations is a fundamental skill. It lets us understand relationships between different quantities. For example, if you know the equation involves a cube root, like in our problem, the first step is to eliminate it. We do this by mathematically manipulating the equation. This often involves performing the same operation on both sides of the equation. Doing so maintains the equality.
  • Identify the operations affecting the variable.
  • Apply opposite operations to simplify the equation.
  • Continue until the variable is isolated.
Understanding how to carefully apply and reverse operations is key to successfully solving equations.
Cube Root
The cube root of a number is another number that, when multiplied by itself twice (three times in total), results in the original number. For instance, the cube root of 8 is 2 because 2 脳 2 脳 2 equals 8. In an equation, cube roots can sometimes complicate the process of solving for a variable, as they are less straightforward than simple roots or squares.
When facing a cube root, remember:
  • To eliminate the cube root, cube both sides of the equation.
  • This "undoes" the cube root, simplifying the expression.
In our example, cubing both sides allowed us to work with a cleaner equation, setting the stage for solving for the desired variable.
Isolating Variables
Isolating a variable means rearranging an equation so that the variable is on one side by itself. This process often involves a combination of basic algebraic operations such as addition, subtraction, multiplication, and division.
In general, to isolate a variable:
  • Identify the operations in the equation.
  • Use inverse operations to "undo" them.
  • Perform the same operation on both sides to maintain balance.
In our original problem, once we eliminated the cube root, we used multiplication and division to finally isolate and solve for V. This step is crucial because it transfers the focus of the equation solely onto the variable we are interested in.
Volume of a Sphere
The volume of a sphere is a measure of the amount of space inside it. Understanding the volume involves both geometry and algebra. The formula for the volume of a sphere is:\[ V = \frac{4}{3}\pi r^3 \]Here, \( r \) represents the radius of the sphere. The formula tells us how the volume changes with the radius. If you know the radius, calculating the volume becomes straightforward.
  • The formula involves the constant \( \pi \), approximately 3.14159.
  • The radius \( r \) is raised to the third power, reflecting the three-dimensional nature of a sphere.
By expressing the volume in terms of \( r \), the formula shines light on the relationship between the size of the sphere and the space it occupies.

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

Simplify completely. Assume all variables represent positive real numbers. $$\frac{4}{\sqrt{w}}$$

Rationalize the denominator of each expression. $$-\frac{20}{\sqrt{8}}$$

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. $$\frac{\sqrt{u}}{\sqrt{u}-\sqrt{v}}$$

Simplify completely. $$\frac{\sqrt{45}+6}{9}$$

Simplify completely. $$\frac{36+20 \sqrt{2}}{12}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Decision 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.