Problem 62 - Calculus with Concepts in Calculus

Question

Express the radius \(r\) of the base of a right circular cone as a function of the volume \(V\) and height \(h\).

Step-by-Step Solution

Answer

The radius \(r\) as a function of \(V\) and \(h\) is \(r = \sqrt{\frac{3V}{\pi h}}\).

Step 1: Understanding the Formula for Cone Volume

The volume of a right circular cone can be expressed with the formula: \[ V = \frac{1}{3} \pi r^2 h \], where \(V\) is the volume, \(r\) is the radius of the base, and \(h\) is the height of the cone.

Step 2: Rearranging the Cone Volume Formula

To express the radius \(r\) as a function of volume \(V\) and height \(h\), first isolate \(r^2\) in the formula. Begin by multiplying both sides of the equation by 3: \[ 3V = \pi r^2 h \].

Step 3: Solving for Radius Square

Now, to solve for \(r^2\), divide both sides by \(\pi h\): \[ r^2 = \frac{3V}{\pi h} \].

Step 4: Calculating the Radius

Take the square root of both sides to solve for \(r\): \[ r = \sqrt{\frac{3V}{\pi h}} \]. Now, \(r\) is expressed as a function of \(V\) and \(h\).

Key Concepts

Right Circular Cone

The right circular cone is a three-dimensional geometric shape. It has a circular base and a pointed top known as the apex. One characteristic that defines a right circular cone is that the line drawn from the apex to the center of the base forms a right angle with the base itself. This unique shape is encountered in everyday life, with objects like ice cream cones and party hats serving as perfect examples.
The right angle created by the height and the circular base is crucial because it helps in accurately calculating the cone's volume. To define a right circular cone:

• The central axis runs perpendicularly through the base to the apex.
• The curve that outlines the cone is smooth and symmetrical.

These properties make the right circular cone a simple, yet fascinating shape to explore in geometry.

Radius as a Function

In the context of this exercise, we need to express the radius of the cone's base, denoted as \(r\), using the volume \(V\) and height \(h\) of the cone. Given the formula for the cone's volume: \[ V = \frac{1}{3} \pi r^2 h \], we can rearrange this equation to isolate \(r\).
To find \(r^2\), multiply both sides by 3 and then divide by \(\pi h\):
\[ 3V = \pi r^2 h \]
\[ r^2 = \frac{3V}{\pi h} \].
Finally, take the square root of both sides to find \(r\):
\( r = \sqrt{\frac{3V}{\pi h}} \).
This equation shows the radius as a function of the cone's volume and height. It allows for calculations of the base radius given any specific measurements of volume and height.

Volume and Height Relationship

The formula for the volume of a cone, \( V = \frac{1}{3} \pi r^2 h \), indicates a direct relationship between the volume, the square of the radius, and the height of the cone. The volume changes with any alterations in these dimensions.
The factor of \(\frac{1}{3}\) in the formula reflects that a cone's volume is one third that of a cylinder with the same base and height, illustrating its unique geometric properties. Here’s how these elements interplay:

• If the height \(h\) increases while keeping the volume constant, the radius must decrease to accommodate this change, and vice versa.
• The cone's volume will become larger as either the radius or height increases, given that the other variables remain constant.

Understanding this relationship is essential for solving problems that involve scaling up or down the dimensions of cones while maintaining their proportions.