Problem 78 - Algebra for College Students

Question

Set up an equation and solve each of the following problems. The total surface area of a right circular cone is \(108 \pi\) square feet. If the slant height of the cone is twice the length of a radius of the base, find the length of a radius.

Step-by-Step Solution

Answer

The radius is 6 feet.

Step 1: Understanding the Problem

We are given a right circular cone with a total surface area of \(108 \pi\) square feet, and we know that the slant height \(l\) is twice the length of the radius \(r\). We need to find \(r\).

Step 2: Surface Area Formula for a Cone

The surface area \(A\) of a right circular cone is given by the formula: \(A = \pi r^2 + \pi r l\), where \(r\) is the radius and \(l\) is the slant height.

Step 3: Substituting Known Values

We know \(A = 108 \pi\) and \(l = 2r\), as per the problem statement. Substitute these values into the surface area formula: \(108 \pi = \pi r^2 + \pi r(2r)\).

Step 4: Simplifying the Equation

Simplify the equation \(108 \pi = \pi r^2 + 2\pi r^2\) by combining like terms: \(108 \pi = 3\pi r^2\).

Step 5: Solving for the Radius

Divide both sides by \(3\pi\): \(108 \pi / 3\pi = r^2\). This simplifies to \(36 = r^2\).

Step 6: Finding the Final Answer

Take the square root of both sides to solve for \(r\): \(r = \sqrt{36}\), which gives \(r = 6\).

Key Concepts

Right Circular Cone

A right circular cone is a three-dimensional geometric shape that is commonly encountered in both mathematics and real-world applications. Its defining characteristics include:

• An upright axis that makes it distinct from oblique cones.
• A circular base that is flat and lies perfectly horizontal.
• A pointed tip, known as the apex, situated directly above the center of the circular base.

The name 'right' in 'right circular cone' comes from the cone's axis being perpendicular to its base. This is analogous to a right angle in a right triangle, where the right angle ensures perpendicularity.
The base radius and the perpendicular axis of a right circular cone are fundamental in calculating various geometric properties, such as volume or surface area. Knowing these properties helps in deriving other measures, which are crucial in real-life scenarios, such as engineering and architecture.
Understanding a right circular cone helps in solving problems regarding its volume, dimensions, and surface area effortlessly, making it a preferable starting point for students to delve into more complex geometry.

Radius of a Cone

The radius of a cone is the distance from the center of the circular base to its edge. It is a critical component when working with cones as it directly affects other measurements:

• The volume of the cone, calculated using the formula \( V = \frac{1}{3} \pi r^2 h \), where \(h\) is the height.
• The surface area, comprising both the base area and the lateral surface area, which requires the radius for standard calculation.

In exercises, like the one given in the textbook, identifying the radius first is crucial as it impacts all subsequent calculations. Given its effect on both the lateral and total surface area formulas, understanding the radius helps students grasp how size changes alter overall dimensions.
In problems where direct measurement isn't available, students might be required to derive the radius using algebraic formulations, as done in the given problem. Familiarity with simplifying equations, like solving for \(x\) in quadratic equations, can be very handy when working with formulas involving the radius.

Slant Height of a Cone

The slant height of a cone is the diagonal distance from the top of the cone's apex down to any point on the edge of the circular base. It is different from the vertical height of the cone and adds an important dimension to understanding the cone's geometry.

• It is always the hypotenuse of a right triangle formed within the cone, with the cone's height and radius forming the other two sides.
• Calculating the slant height (denoted as \(l\)) is necessary to determine the lateral surface area of the cone, given by the formula \( \pi r l \).

In many problems, you will find the relationship between the slant height and radius crucial, especially when the lateral surface area must be determined. For example, in the provided exercise, the slant height was given as twice the radius, a key relationship that simplified the problem-solving process.
Understanding how the slant height compares to the other dimensions of the cone can greatly help in visualizing the cone and in intricate calculations, particularly in cases involving compound shapes with conical components.