Problem 55 - Calculus of a Single Variable

Question

Lateral Surface Area of a Cone A right circular cone is generated by revolving the region bounded by \(y=3 x / 4\) , \(y=3,\) and \(x=0\) about the \(y\) -axis. Find the lateral surface area of the cone.

Step-by-Step Solution

Answer

The lateral surface area of the cone is \(12π\) square units

Step 1: Identify the radius and the slant height of the cone

The mentioned region represents a right triangle, and when revolved around the y-axis, it forms a cone. In the shear case, the peak of the triangle represents the apex of the cone and the x-intercept of the line \(y=3x/4\) gives the radius (\(r\)) of the cone while the height of the triangle provides the slant height (\(l\)) of the cone. Make \(y=3x/4\) equal with \(y=3\), to find \(r=4\). Also, from the equation \(y = 3\), it defines that the height of the triangle is 3, so \(l=3\).

Step 2: Apply the lateral surface area formula of the cone

The formula to find the lateral surface area of a cone is \(πrl\). Substitute \(r\) and \(l\) from Step 1 into this formula: \(π*4*3 = 12π\) square units.

Key Concepts

Right Circular Cone

A right circular cone is a three-dimensional geometric shape that is formed by a circular base and a point called the apex or vertex, not lying on the base. The apex is connected to the perimeter of the base by a smooth surface. This shape resembles an ice cream cone or a party hat.

• The base of a right circular cone is a perfect circle.
• The line segment from the apex perpendicular to the base is called the height.
• When a cone is referred to as "right," it means that its apex is exactly above the center of the base, forming a right angle with the base.

Understanding the properties of a right circular cone helps in calculating various dimensions like the slant height and the lateral surface area.

Revolution About the y-axis

Revolution about the y-axis is a process often used in calculus and geometry to generate three-dimensional shapes. When a two-dimensional shape is revolved around the y-axis, it sweeps out a three-dimensional object.

• This method is commonly used for generating solids of revolution, such as cones and spheres.
• In this scenario, revolving a triangle about the y-axis creates a cone. The triangle's vertical side aligns with the y-axis, serving as the central axis of the cone.

This concept is crucial when analyzing or designing objects like vases or funnels, where the symmetry around a central axis is desired.

Lateral Surface Area Formula

The lateral surface area of a cone is the area of the cone's surface excluding the base. This area forms the curved part of the cone, known as the lateral surface.

• The formula to find this surface area is \( \pi r l \), where \( r \) is the radius of the base and \( l \) is the slant height.
• The lateral surface area is important in practical applications, like determining the amount of material needed to cover a conical tent or constructing a funnel.

In the exercise, using the values \( r = 4 \) and \( l = 3 \), the lateral surface area is calculated as \( 12\pi \) square units, showcasing the simplicity and usefulness of the formula.

Slant Height of a Cone

The slant height of a cone is the distance from the apex of the cone to any point on the edge of the base. It forms a diagonal along the cone's surface and is crucial for calculating the lateral surface area.

• This height is part of the right triangle formed by the height of the cone, the radius of the base, and the cone itself.
• It can be found using the Pythagorean theorem if the height and radius of the cone are known.

In geometry and practical applications, knowing the slant height allows for accurate measures and constructions involving cones, such as making precise costumes or engineering components.