Problem 110 - College Algebra with Modeling and Visualization

Question

Geometry The surface area of a cone (excluding the bottom) is given by \(S=\pi r \sqrt{r^{2}+h^{2}},\) where \(r\) is its radius and \(h\) is its height, as shown in the figure. If the height is twice the radius, write a formula for \(S\) in terms of \(r\) (GRAPH CANT COPY)

Step-by-Step Solution

Answer

The formula for surface area in terms of \( r \) is \( S = \pi r^2 \sqrt{5} \).

Step 1: Substitute the Height

Since the height is twice the radius, we have \( h = 2r \). Substitute \( h \) into the surface area formula for the cone. The original formula is \( S = \pi r \sqrt{r^2 + h^2} \). After substitution, it becomes \( S = \pi r \sqrt{r^2 + (2r)^2} \).

Step 2: Simplify the Expression

Inside the square root, expand and simplify \( (2r)^2 \). This results in \( (2r)^2 = 4r^2 \). Replace \( h^2 \) with \( 4r^2 \) in the formula. Now, the expression inside the square root is \( r^2 + 4r^2 \).

Step 3: Combine Like Terms

Combine the terms inside the square root: \( r^2 + 4r^2 = 5r^2 \). The formula now is \( S = \pi r \sqrt{5r^2} \).

Step 4: Simplify the Square Root

Calculate the square root of \( 5r^2 \). This becomes \( \sqrt{5} \cdot r \), since \( \sqrt{5r^2} = \sqrt{5} \cdot \sqrt{r^2} = \sqrt{5} \cdot r \).

Step 5: Write the Final Formula

Substitute \( \sqrt{5} \cdot r \) back into the surface area formula to get \( S = \pi r (\sqrt{5} \cdot r) = \pi r^2 \sqrt{5} \).

Key Concepts

surface area of a cone

Understanding the surface area of a cone is essential in geometry, as it helps to calculate the amount of material needed for crafting or constructing conical objects. The formula for calculating the surface area (excluding the base) is \( S = \pi r \sqrt{r^2 + h^2} \). Here:

• \( r \) is the radius of the cone's base.
• \( h \) is the vertical height of the cone.

This formula accounts for the slant height of the cone, found through the Pythagorean theorem. When we apply this formula, we find the curved surface area—the actual area you would paint or wrap, for example. By manipulating this formula, as in the exercise, it helps us determine how changing one dimension (like the height) affects the overall surface area.

mathematical formulas

Formulas are cornerstones in mathematics, offering a structured way to solve problems. In geometry, formulas express relationships between different elements. For the cone's surface area, the formula \( S = \pi r \sqrt{r^2 + h^2} \) is derived from understanding how a cone's physical dimensions relate to its surface area.
This specific formula:

• Incorporates basic geometric shapes like circles and right triangles.
• Utilizes the concept of a slant height as part of its calculation.
• Demonstrates how extended mathematical concepts, such as the Pythagorean theorem, are integrated into geometry.

For those learning math, practicing substituting and rearranging formulas is a crucial skill. It helps build confidence in recognizing and applying them to new problems, much like in the exercise where the height is replaced with twice the radius.

simplifying expressions

Simplifying expressions is a fundamental aspect of solving mathematical problems. It involves reducing complex expressions to their simplest forms, which makes calculations easier and results clearer. In this exercise, we simplify the formula step-by-step. First, notice the substitution of \( h = 2r \). This is possible because the problem states the height is twice the radius.
The simplification then follows these steps:

• Replace \( h \) with \( 2r \), changing the formula to \( S = \pi r \sqrt{r^2 + (2r)^2} \).
• Expand \( (2r)^2 \) into \( 4r^2 \).
• Combine like terms inside the square root to get \( 5r^2 \).
• Simplify the square root as \( \sqrt{5} \cdot r \).

These steps put the expression in its simplest form: \( S = \pi r^2 \sqrt{5} \). By mastering these simplification techniques, tackling a wide range of math problems becomes more intuitive.