Learning Materials
- Explanations
Explanations
- Biology
- Business Studies
- Chemistry
- Chinese
- Combined Science
- Computer Science
- Economics
- Engineering
- English
- English Literature
- Environmental Science
- French
- Geography
- German
- History
- Human Geography
- Italian
- Law
- Macroeconomics
- Marketing
- Math
- Microeconomics
- Nursing
- Physics
- Politics
- Psychology
- Sociology
- Spanish
- Textbooks
Textbooks
- Biology
- Business Studies
- Chemistry
- Combined Science
- Computer Science
- Economics
- English
- Environmental Science
- Geography
- History
- Math
- Physics
- Psychology
- Sociology
Features
- Flashcards
- Vaia IA
- Notes
- Study Plans
- Study Sets
- Exams
Discover
- Magazine
- Find a job
- Mobile App
Chapter 3: Problem 15
The volume \(V\) of a right circular cylinder varies jointly as the height \(h\)of the cylinder and as the square of the radius \(r\) of the cylinder.
Short Answer
Expert verified
The volume formula is \( V = \pi r^2 h \).
Step by step solution
01
Identify the Relationship
The problem states that the volume of a right circular cylinder varies jointly as the height and the square of the radius. This means the volume is proportional to the height and the square of the radius.
02
Set Up the Joint Variation Equation
When one quantity varies jointly as two others, it can be expressed as a constant times the product of those two quantities. Therefore, we can write: \[ V = k \times h \times r^2 \] where \(k\) is the constant of proportionality.
03
Volume Formula of a Cylinder
For a right circular cylinder, the volume formula is generally given by:\[ V = \pi r^2 h \]Here, the constant \(k\) corresponds to \(\pi\).
04
Write the Final Equation
Thus, the equation representing the relationship is:\[ V = \pi r^2 h \]
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
right circular cylinder
A right circular cylinder is a three-dimensional geometric shape. It consists of two parallel circular bases connected by a curved surface. The axis of the cylinder is the line segment connecting the centers of the circular bases, and it is perpendicular to the bases. The right circular cylinder is a common object in geometry, engineering, and daily life, such as in the shape of a can or a pipe. To visualize it better, think of a straight column with perfectly round and flat ends.
volume formula
The volume of a right circular cylinder measures the amount of space inside the cylinder. To find this volume, you use the formula: \[ V = \pi r^2 h \]. Here's a breakdown of the formula:
- The symbol \( \pi \) (pi) is a mathematical constant, approximately equal to 3.14159.
- The variable \( r \) represents the radius of the cylinder's base.
- The variable \( h \) represents the height of the cylinder.
When these components are multiplied together, they give the volume. This formula captures the three-dimensional nature of the object, combining the area of the base (\( \pi r^2 \)) and the height (\( h \)).
proportionality constant
In the context of joint variation, a proportionality constant is a value that relates two or more variables in an equation. For a right circular cylinder, the volume \( V \) is related to the height \( h \) and the square of the radius \( r^2 \). The equation \( V = k h r^2 \) includes a proportionality constant, represented by \( k \). In this case, \( k \) corresponds to \( \pi \), simplifying the equation to the familiar volume formula \( V = \pi r^2 h \). This constant helps in converting geometric relationships into an actual numerical value for calculations.
joint variation
Joint variation describes how one quantity varies directly with the product of two or more other quantities. In this exercise, the volume \( V \) of the right circular cylinder varies jointly with the height \( h \) and the square of the radius \( r^2 \). This means that if you increase either the height or the radius, the volume will also increase proportionally. Mathematically, we express this relationship as \( V = k h r^2 \). Here, \( k \) is the constant of proportionality. Understanding joint variation helps in designing and calculating real-world objects by considering how multiple factors influence outcomes.
One App. One Place for Learning.
All the tools & learning materials you need for study success - in one app.
Get started for free
![Problem 15 The volume \(V\) of a right circ... [FREE SOLUTION] (31)](https://i0.wp.com/website-cdn.studysmarter.de/sites/14/2023/12/Rectangle-22721-2.png "Problem 15 The volume \(V\) of a right circ... [FREE SOLUTION] (31)")
Most popular questions from this chapter
Recommended explanations on Math Textbooks
Decision Maths
Read ExplanationMechanics Maths
Read ExplanationApplied Mathematics
Read ExplanationDiscrete Mathematics
Read ExplanationPure Maths
Read ExplanationProbability and Statistics
Read ExplanationWhat do you think about this solution?
We value your feedback to improve our textbook solutions.
Study anywhere. Anytime. Across all devices.
Sign-up for free
This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept
Privacy & Cookies Policy
Privacy Overview
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.
Always Enabled
Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.
Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.
![Problem 15 The volume \(V\) of a right circ... [FREE SOLUTION] (2024)](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mP8/h8AAvMB+NzmkbcAAAAASUVORK5CYII=)