Problem 38 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

The height of a right circular cone is one third of the diameter of the base. (a) Express its volume as a function of its height, \(h\). (b) Express its volume as a function of \(r\), the radius of its base.

Step-by-Step Solution

Answer

(a) The volume of the cone as a function of \( h \) is \( V = \frac{3}{4}\pi h^3 \). (b) The volume of the cone as a function of \( r \) is \( V = \frac{2}{9}\pi r^3 \)

Step 1 - Express the Diameter in Terms of \( h \)

Given the height of a right circular cone is one third of the diameter of the base, we write the diameter \( d \) in terms of \( h \). So, \( d = 3h \)

Step 2 - Express Volume as a Function of \( h \)

Since the diameter \( d \) is twice the radius, we substitute \( r \) with \( \frac{d}{2} = \frac{3h}{2} \) in the volume formula, obtaining \( V = \frac{1}{3} \pi \left(\frac{3h}{2}\right)^2h \). After simplification, the formula becomes \( V = \frac{9}{12}\pi h^3 = \frac{3}{4}\pi h^3 \)

Step 3 - Express Volume as a Function of \( r \)

We substitute \( h \) with \( \frac{2r}{3} \) in the volume formula, producing \( V = \frac{1}{3} \pi r^2(\frac{2r}{3}) \). After simplification, the formula becomes \( V = \frac{2}{9}\pi r^3 \)

Key Concepts

Height and Diameter Relationship

In the context of geometry, understanding the relationship between the height and diameter of a cone is crucial. When we talk about a right circular cone, the height is perpendicular from the base to its apex. In this scenario, the given height is one third of the diameter of the base. This means that if you know the diameter, you can quickly find the height and vice versa, by utilizing simple multiplication or division.
The formula given in the exercise is:

• For diameter, if height is given: \(d = 3h\)
• For height, if diameter is given: \(h = \frac{d}{3}\)

This relationship helps in setting up the equations correctly when calculating the cone's volume.

Volume Formula

The volume of a cone is the amount of space enclosed by it, and it can be calculated using its height and radius of the base. The standard volume formula for a cone is given by:

• \( V = \frac{1}{3} \pi r^2 h \)

This formula uses three key elements:

• \(\pi\): A constant, approximately 3.14159, that relates to the circle's circumference and area.
• \(r^2\): The square of the radius of the base.
• \(h\): The height of the cone.

Plug these values into the formula to find the cone's volume. Remember, finding the volume means calculating how much space is inside the cone, which is usually expressed in cubic units.

Right Circular Cone

A right circular cone is defined by having a circular base and a vertex or apex that is aligned above the center of the base. This makes the perpendicular distance from the base to the apex the height of the cone. This shape comes into play when dealing with many mathematical problems and real-world applications such as designing funnels or ice cream cones.
Key characteristics include:

• The base is circular
• The height is perpendicular to the base
• The vertex is directly above the base's center

Understanding these properties aids in deriving important formulas and equations related to cones, ensuring accurate mathematical solutions.

Radius of a Circle

The radius is an important metric in understanding and calculating the dimensions of a cone’s base. It is defined as the distance from the center of a circle to any point on its circumference. Knowing the radius allows calculation of other crucial factors such as the diameter and the area of the circle.
In the case of cones, it also plays a significant part in determining the volume. The radius can often be denoted as \(r\) in equations. Here’s a quick breakdown:

• The diameter \(d\) is twice as long as the radius: \(d = 2r\)
• The area of the circle is expressed as \( \pi r^2 \)

In volume calculations, the square of the radius is very significant because it directly affects the overall volume of the cone, making understanding and correctly calculating the radius vital for accurate results.