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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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