Problem 47 - Calculus: Early Transcendentals

Question

Find the volume of the described solid \( S \). A right circular cone with height \( h \) and base radius \( r \).

Step-by-Step Solution

Answer

The volume of the cone is \( V = \frac{1}{3} \pi r^2 h \).

Step 1: Understanding the Problem

We need to find the volume of a right circular cone, which is a 3-dimensional geometric shape with a circular base and a pointed top. The cone is described by its height \( h \) and the radius of its base \( r \).

Step 2: Formula for the Volume of a Cone

The volume \( V \) of a cone is given by the formula: \( V = \frac{1}{3} \pi r^2 h \), where \( r \) is the radius of the base and \( h \) is the height of the cone.

Step 3: Substitute Given Values

Substitute the given values of the radius \( r \) and height \( h \) into the formula to calculate the volume. If specific values for \( r \) and \( h \) are given, use those in this step, otherwise, continue with the variables.

Step 4: Simplify the Expression

After substituting the values, perform the necessary arithmetic operations and simplify the expression to find the volume. This will usually involve multiplying \( \pi \), squaring the radius, multiplying by the height, and then dividing by 3.

Key Concepts

Geometry

Geometry is the branch of mathematics that focuses on the properties and relationships of points, lines, surfaces, and solids. It helps us understand and describe the physical shapes around us. In geometry, a cone is one of the many 3-dimensional shapes you might encounter, characterized by its round base and a single vertex. This subject allows us to explore how different shapes relate to each other in space and enables the calculation of various properties, such as area and volume, that define these shapes.

Cone

A cone is a 3-dimensional geometric shape that tapers smoothly from a flat circular base to a point called the vertex or apex. It resembles a party hat or an ice cream cone. There are various types of cones, but the most common is the right circular cone.

• Right circular cone: The axis of the cone is perpendicular to the base.
• Slant height: The distance from the base to the vertex along the outside of the cone.

Cones are commonly found in both natural and man-made structures, and understanding them is useful in fields ranging from architecture to astrophysics.

Volume Calculation

Calculating the volume of a cone involves determining how much space the cone occupies. Volume calculation helps us to quantify the capacity within a 3D object. For a cone, understanding the relationship between its height and base is crucial for arriving at its volume.

• The base is a circle with radius \( r \).
• The height \( h \) is the perpendicular distance from the base to the vertex.

When you encounter a cone, these are the parameters you need to measure in order to perform a volume calculation.

Mathematical Formula

The formula to find the volume of a cone is essential knowledge in mathematics. This formula captures the geometric properties of the cone and expresses them in a way that allows you to find how much space the cone occupies. The formula used is: \[V = \frac{1}{3} \pi r^2 h \]Here's a breakdown of this formula:

• \( V \) is the volume.
• \( \pi \approx 3.14159 \), a constant that represents the ratio of the circumference of a circle to its diameter.
• \( r^2 \) is the radius squared, which means you multiply the radius by itself.
• \( h \) is the height.

This formula shows that the volume of a cone is one-third of the volume of a cylinder with the same base and height, which is an interesting mathematical fact.