Problem 60 - Algebra and Trigonometry

Question

Area of a Sphere The surface area \(S\) of a sphere is a function of its radius \(r\) given by $$S(r)=4 \pi r^{2}$$ (a) Find \(S(2)\) and \(S(3)\) . (b) What do your answers in part (a) represent?

Step-by-Step Solution

Answer

S(2) = 16\pi; S(3) = 36\pi; These are the surface areas for radii of 2 and 3.

Step 1: Understand the Formula

The surface area \(S\) of a sphere with radius \(r\) is found using the formula \(S(r) = 4 \pi r^{2}\). We need to use this formula to find \(S(2)\) and \(S(3)\).

Step 2: Calculate S(2)

Plug \(r = 2\) into the surface area formula: \[S(2) = 4 \pi (2)^{2} = 4 \pi \times 4 = 16 \pi\].

Step 3: Calculate S(3)

Plug \(r = 3\) into the surface area formula: \[S(3) = 4 \pi (3)^{2} = 4 \pi \times 9 = 36 \pi\].

Step 4: Interpret the Results

The results \(S(2) = 16 \pi\) and \(S(3) = 36 \pi\) represent the surface areas of spheres with radii 2 and 3, respectively.

Key Concepts

Understanding a Sphere

A sphere is a perfectly symmetrical three-dimensional shape, just like a ball or a globe. It is defined in geometry as a set of points that are equidistant from a central point.

Let's break down some key attributes of a sphere:

• It has no edges or vertices.
• Every point on the surface is equally distant from the center.
• Its surface is continuous and smooth, making it an object of symmetry.

These attributes make a sphere both interesting and unique in geometry studies, providing an ideal shape to explore concepts such as surface area.

Exploring the Radius

The radius of a sphere is a straight line extending from the center to any point on its surface. This measurement is crucial as it helps define the size of the sphere, influencing calculations of surface area and volume.

To better understand the radius, consider:

• It is always half the length of the diameter, which is the longest line you can draw through the sphere passing through the center.
• The radius determines the surface area and volume. A larger radius results in a larger sphere, leading to increased surface area and volume.

When solving problems related to spheres, knowing the radius simplifies using formulas such as the one to calculate surface area.

The Language of Geometry

Geometry, the branch of mathematics dealing with shapes and space, brings us the sphere as an essential object of study. It provides tools and methods to calculate properties like area and volume.

In geometry:

• Figures like spheres, cylinders, and cubes help us understand and visualize space.
• Mathematical reasoning extends our comprehension of form and space interactions.

Geometry forms the basis of many real-world applications, from designing video games to constructing buildings. A key competency is understanding formulas and how dimensions such as radius factor into geometric calculations.

Mathematical Calculations Made Simple

Mathematics isn't just about numbers; it's about understanding patterns and solving problems. Sphere-related problems often involve calculating the surface area and volume.

Here's how you can approach these with confidence:

• Use formulas, like the surface area formula for spheres: \( S(r) = 4 \pi r^{2} \).
• Substitute known values for variables accurately, as demonstrated with finding \( S(2) \) and \( S(3) \).
• Understand what results represent – in this case, the spatial extent of the sphere's surface.

With practice, recognizing the context and meaning behind calculations strengthens your mathematical skills. This empowers you to tackle complex problems more effectively.