Problem 70 - 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

\(S(2) = 16\pi\) and \(S(3) = 36\pi\); they represent the surface areas of spheres with radii 2 and 3.

Step 1: Understand the Surface Area Formula

The formula for the surface area of a sphere is given by \( S(r) = 4 \pi r^2 \). This means for any sphere, if you know the radius, \( r \), you can calculate the surface area, \( S \), by plugging \( r \) into this formula.

Step 2: Calculate \( S(2) \)

To find \( S(2) \), substitute \( r = 2 \) into the surface area formula: \[S(2) = 4 \pi (2)^2 = 4 \pi \times 4 = 16 \pi.\]

Step 3: Calculate \( S(3) \)

Next, calculate \( S(3) \) by substituting \( r = 3 \) into the 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 of 2 units and 3 units, respectively. Each value is expressed in terms of \( \pi \).

Key Concepts

Understanding the Radius of a Sphere

The radius of a sphere is a crucial measurement defining the size of the sphere from its center to any point on its surface. It's represented by the letter \( r \). This single measurement can help determine several other key characteristics of a sphere, including its volume and surface area.
Remember, every point on the surface of a sphere is equidistant from its center, which underlines why the radius is so important.
A change in the radius affects the sphere's overall dimensions significantly, influencing both its surface area and the space it occupies. Therefore, understanding the radius is the first step in exploring the geometry of spheres.
In exercises involving spheres, you often calculate these attributes to understand better the shape's properties. Let's see how mathematical formulas incorporate the radius.

Deciphering Mathematical Formulas in Spheres

Mathematical formulas are essential for unlocking the properties of geometric shapes such as spheres. The formula for the surface area "\( S \)" of a sphere is:

• \( S(r) = 4 \pi r^2 \)

This formula shows that the surface area is directly related to the square of its radius.
\( \pi \), a constant approximately equal to 3.14159, represents the ratio of the circumference of any circle to its diameter and plays a pivotal role in calculations involving circles and spheres.
To find the surface area with a given radius, you simply plug the radius into the formula:

• Example: For a radius \( r = 2 \):
• \( S(2) = 4 \pi (2)^2 = 16 \pi \)
• For a radius \( r = 3 \):
• \( S(3) = 4 \pi (3)^2 = 36 \pi \)

The presence of \( r^2 \) in the formula highlights that any increase in the radius increases the surface area exponentially, rather than linearly, showcasing how geometry scales in three-dimensional spaces.

Interpreting Mathematical Results: Surface Area of Spheres

When you calculate the surface area using the formula \( S(r) = 4 \pi r^2 \), the results have both theoretical and practical implications.
They give us a concrete understanding of the sphere's external size. In our exercise, the calculations \( S(2) = 16 \pi \) and \( S(3) = 36 \pi \) tell us the sizes of spheres with radii of 2 and 3 units, respectively.

• Each result is an area measurement, broadly in terms of "square units," decisive for determining how much space the surface covers.
• Expressing the answers with \( \pi \) keeps them precise, avoiding unnecessary rounding.

Understanding this aspect is vital in real-world contexts, like material science, where knowing the exact surface area can affect the cost and feasibility of manufacturing spherical objects. The interpretation of these mathematical results bridges abstract concepts with tangible applications, underscoring the real-world importance of learning geometry.