Problem 61 - Calculus of a Single Variable

Question

Using a Cone A cone of height \(H\) with a base of radius \(r\) is cut by a plane parallel to and \(h\) units above the base, where \(h

Step-by-Step Solution

Answer

The volume \( V_F \) of the Frustum is given by \( V_F = \frac{1}{3} \pi r^2 H - \frac{1}{3} \pi r^2 \cdot (\frac{(H - h)}{H})^2 (H - h) \).

Step 1: Finding the radius of the smaller Cone

From similar triangles, the radius \( R \) of the smaller cone is given by \( R = r \cdot \frac{(H - h)}{H} \).

Step 2: Calculating the volume of the larger Cone

The volume \( V_L \) of the original, larger Cone is given by \( V_L = \frac{1}{3} \pi r^2 H \).

Step 3: Calculating the volume of the smaller Cone

The volume \( V_S \) of the smaller Cone is given by \( V_S = \frac{1}{3} \pi R^2 (H - h) \), substituting \( R \) from step 1, we get \( V_S = \frac{1}{3} \pi r^2 \cdot (\frac{(H - h)}{H})^2 (H - h) \).

Step 4: Calculating the volume of the Frustum

The volume \( V_F \) of the Frustum is the difference between the larger and smaller Cone, \( V_F = V_L - V_S = \frac{1}{3} \pi r^2 H - \frac{1}{3} \pi r^2 \cdot (\frac{(H - h)}{H})^2 (H - h) \). Simplifying this will give the volume of the Frustum.

Key Concepts

Understanding Similar Triangles in a Cone

When discussing the frustum of a cone, we encounter a concept of similar triangles. Imagine a large cone split by a parallel plane, forming a smaller cone and a frustum. Here, the cross-sectional slice through the cone reveals two similar triangles. These triangles share angles and therefore have proportional sides.
For example, if a cone has a height of \( H \) and a plane cuts through it at height \( h \), the remaining heights, \( H-h \) and the two radii (\( R \) for the smaller cone and \( r \) for the larger one) form pairs of these similar triangles.

• By the principles of similar triangles, \( \frac{R}{r} = \frac{(H-h)}{H} \).
• This ratio helps us determine that \( R = r \cdot \frac{(H-h)}{H} \).

This knowledge of similar triangles is crucial for further calculations in the geometry of frustums.

Calculating the Volume of a Cone

The volume of a cone can be found using a straightforward formula derived from its geometric properties. The cone's volume depends on its height and the radius of its base.
The formula for the volume \( V \) of a cone is:

• \( V = \frac{1}{3} \pi r^2 H \)

In this formula:

• \( r \) is the radius of the base.
• \( H \) is the height measured from the base to the tip.
• \( \pi \) is a mathematical constant approximately equal to 3.14159.

This formula helps us understand how the three-dimensional space occupied by a cone can be measured, a key concept in solving many geometry problems.

Exploring Basic Geometry of a Cone

Geometry of conical shapes introduces interesting challenges and solutions. The cone itself is a three-dimensional object with a circular base that tapers to a point called the apex.
Key features of cone geometry include:

• **Base**: The circular surface at the bottom.
• **Apex**: The pointed top of the cone.
• **Slant Height**: The diagonal distance from the apex to any point on the edge of the base.

The height of a cone is always taken as a perpendicular line from the apex to the center of the base. Understanding these elements is vital for understanding more complex features of conical geometry, such as frustums.

Mastering Volume Calculation for a Frustum

Calculating the volume of a frustum involves understanding how a cone is sectioned by a plane slice parallel to the base. The resulting frustum is the portion below the slice.
To find the volume of the frustum \( V_F \), follow these steps:

• First, calculate the volume of the original larger cone using \( \frac{1}{3} \pi r^2 H \).
• Then, determine the volume of the smaller cone formed above the slice using \( V_S = \frac{1}{3} \pi \left(r\cdot \frac{(H-h)}{H}\right)^2 (H-h) \).
• The volume of the frustum is computed as the difference: \( V_F = V_L - V_S \).

Understanding this calculation allows you to measure and compare volumes in complex geometry, an essential skill in mathematics.