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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.
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