Question

A cone has a height of 10 meters and a base with a radius of 3 meters. Find the volume of the cone.

Short answer

The volume of the cone is \(30\pi\) or approximately 94.25 cubic meters.

Step-by-step solution

Understand the Problem

We are given a cone with specific dimensions: the height is 10 meters and the radius of the base is 3 meters. Our task is to find the volume of the cone.

Recall the Formula for the Volume of a Cone

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

Substitute the Given Values into the Formula

Using the formula \( V = \frac{1}{3}\pi r^2 h \), substitute \( r = 3 \) meters and \( h = 10 \) meters: \[ V = \frac{1}{3}\pi (3)^2 (10) \]

Perform the Calculations

First, calculate \( r^2 \) which is \( 3^2 = 9 \). Then, substitute this back into the equation:\[ V = \frac{1}{3}\pi (9)(10) \]Simplify the equation:\[ V = \frac{1}{3}\pi (90) \]Next, calculate \( 90 \div 3 = 30 \):\[ V = 30\pi \]

Calculate the Volume in Exact Terms

The volume in terms of \( \pi \) is \( 30\pi \) cubic meters. If an approximate numerical value is needed, use \( \pi \approx 3.14159 \) to calculate:\[ V \approx 30 \times 3.14159 = 94.2477 \text{ cubic meters} \]

Key concepts

Geometry Formulas

To find the volume of a cone, there's a specific formula that you need to use:

• It is mathematically expressed as: \( V = \frac{1}{3}\pi r^2 h \).
• This formula represents the relationship between the volume \( V \), the base's radius \( r \), and the height \( h \) of the cone.
• The term \( \frac{1}{3} \) implies that the volume of a cone is one-third of the volume of a cylinder with the same base radius and height.
• The \( \pi r^2 \) part covers the area of the circle at the base of the cone.

Using this formula ensures you can calculate the volume of any cone given the necessary measurements of the radius and height. This formula aids in various practical applications, not just academic purposes, including engineering and everyday problem-solving.
One should remember that understanding and correctly applying geometric formulas is foundational for solving a wide range of problems involving three-dimensional objects.

Mathematical Calculations

When calculating the volume of a cone, mathematics comes into play with simple yet crucial steps. First, square the radius:

• For our cone, with a radius of 3 meters, the calculation involves finding \((3)^2\), which equals 9.
• This squared radius is the first essential part of finding the base area of the cone.

After calculating \( r^2 \), multiply this value with the height:

• The height here is 10 meters and so, \( 9 \times 10 = 90 \).
• Then, incorporate the factor of one-third from the volume formula: \( \frac{90}{3} = 30 \).

Multiply this result by \( \pi \) to reach the volume in its exact form:

• Therefore, \( V = 30\pi \), which is sometimes left as the final answer in terms of \( \pi \).
• If a decimal approximation is needed, multiply \( 30 \times 3.14159 \) as done in the calculations to obtain approximately \( 94.2477 \) cubic meters.

These calculations demonstrate basic arithmetic steps such as squaring and multiplying, which are foundational skills in mathematics. They also underline the precision needed in mathematical computations, especially when involving \( \pi \).

Solid Geometry

Solid geometry involves studying three-dimensional shapes like cones, cylinders, and spheres. A cone is a solid with a circular base and a single apex point.

• The specific properties of a cone, such as its height and radius, are crucial for solving volume-related problems.

Understanding a cone's structure helps visualize how it occupies space and how the formula for its volume relates to its physical dimensions:

• The base area is defined by the circle at the bottom of the cone, found using \( \pi r^2 \).
• The volume is then determined by factoring in the height and the unique one-third aspect of the cone's shape relative to a cylinder.

Solid geometry is applicable in many real-world scenarios where volume measurements are essential, such as calculating material quantities needed for construction projects or determining storage capacities. These principles extend well beyond cones to other geometric solids, each with their own specific formulas for volume and surface area calculations.