Problem 59 - Algebra 1: Concepts and Skills

Question

Use the following information. The lateral surface area \(S\) of a cone whose base has radius \(r\) can be found using the formula $$S=\pi \cdot r \sqrt{r^{2}+h^{2}}$$ where h is the height of the cone. Find the lateral surface area of a cone that has a height of 30 centimeters and whose base has a radius of 14 centimeters.

Step-by-Step Solution

Answer

The lateral surface area of the cone is approximately 138.07 cm².

Step 1: Understand the given values and the formula

We are given a height \(h=30cm\) and a radius \(r=14cm\). The formula to calculate the lateral surface area of a cone is \(S=\pi \cdot r \sqrt{r^{2}+h^{2}}\)

Step 2: Substitute the values into the formula

Substitute \(h=30\) and \(r=14\) into the formula: \(S=\pi \cdot 14 \sqrt{14^{2}+30^{2}}\)

Step 3: Computation

Now, perform the arithmetic operations: \(S=\pi \cdot 14 \sqrt{196+900} = \pi \cdot 14 \sqrt{1096}\). Slightly simplifying further, we get \(S= 43.963\pi \,cm^{2}\) after square rooting 1096.

Step 4: Final Calculation

Now multiply this value by \(\pi\) (approximated to 3.1416 for calculation purposes) to get the final result. Thus, \(S = 43.963 \times 3.1416 = 138.07 cm^{2}\) (rounded to two decimal places).

Key Concepts

Lateral Surface Area

The lateral surface area of a cone is the part of the entire surface area that only includes the sides but not the base. This portion of the surface resembles a conical frustum when flattened out. Calculating the lateral surface area helps understand the material needed to form the cone's sides without covering the bottom. Given a cone's base radius and height, the lateral surface area can be determined using specific formulas. This measurement is essential in various real-world applications, such as determining the amount of paint required for coating or crafting a material into a conical shape.

Cone

A cone is a three-dimensional geometric shape. It features a circular base connected to a single vertex or apex. Visually, it looks similar to an ice cream cone or a party hat. It is characterized by its height (the vertical distance from the base to the peak) and the radius of its base. Understanding these dimensions is crucial for calculating other properties of the cone, such as its volume and surface area. In mathematical exercises, knowing how to handle its geometry is important for solving problems relating to cone measurements.

Mathematical Formulas

Mathematical formulas are crucial tools that give a structured approach to solving problems by providing a clear relation among different quantities. For finding the lateral surface area of a cone, the formula used is \[ S = \pi \cdot r \sqrt{r^{2} + h^{2}} \] where:

• \( S \) is the lateral surface area.
• \( r \) is the radius of the cone's base.
• \( h \) is the height of the cone.

By substituting the known values of a cone's dimensions into this formula, the lateral surface area can be easily calculated. Formulas simplify complex calculations and help in achieving accurate results consistently.

Arithmetic Operations

Arithmetic operations involve basic calculations such as addition, subtraction, multiplication, and division. While finding the lateral surface area of a cone, such operations are pivotal. After plugging the known radius and height values into the formula, the next step involves these arithmetic steps:

• Square the radius and the height, respectively.
• Add these squared values.
• Find the square root of the sum.
• Multiply the resulting square root by \( \pi \) and the radius.

Performing accurate arithmetic operations is essential to ensure the correctness of your mathematical results. Mastery of these basic operations can significantly simplify solving more complex problems.