Question

The surface area \(S\) of a sphere with radius \(r\) is shown in the table. Write a model that best fits the data. $$\begin{array}{|l|c|c|c|c|c|}\hline \text { Radius, } r & 1 & 2 & 3 & 4 & 5 \\\\\hline \text { Surface area, } s & 4 \pi & 16 \pi & 36 \pi & 64 \pi & 100 \pi \\\\\hline\end{array}$$

Short answer

The model that best fits the given data is \(s = 4 \pi r^2\).

Step-by-step solution

Identify the Pattern in the Data

By observing the data in the table, it becomes apparent that, as the radius doubles, the surface area quadruples, confirming that the surface area is proportional to the square of the radius. This observation gives us the first part of the model needed.

Test the Pattern

To confirm the identified pattern, plug in the given radius values into the model \(s = 4 \pi r^2\) and check whether the resultant values for \(s\) match those given in the table. This will help confirm whether the model fits the provided data.

Finalize the Model

Given that the model \(s = 4 \pi r^2\) correctly calculates the surface area for all the given radius values, this model can be confirmed as the correct model for the given data.

Key concepts

Geometric Modeling

Geometric modeling involves creating a mathematical framework to represent shapes and forms in space. In the context of a sphere's surface area, we use geometric modeling to derive a formula that can accurately calculate the surface area based on the sphere’s radius. The process begins with recognizing the shape's properties; for instance, a sphere is perfectly symmetrical, and its surface area at any given point is a function of its radius.

The formula for the surface area of a sphere, as derived from geometric principles, is given by the equation \( S = 4\pi r^2 \), where \( S \) represents the surface area and \( r \) represents the radius. This model is a simple yet powerful representation of the relationship between a sphere's surface area and its radius, providing a practical tool for solving various problems in mathematics and physics.

Applying Geometric Modeling to Real-World Problems

Geometric models, such as the sphere's surface area formula, are not just theoretical explorations but have practical applications in fields like architecture, engineering, and computer graphics. These models help in designing spherical objects ranging from sports equipment to planetary bodies.

Quadratic Relationships

Quadratic relationships feature prominently in various aspects of algebra and are characterized by an equation of the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The surface area of a sphere exhibits a quadratic relationship with its radius, as seen in the equation \( S = 4\pi r^2 \)—a representation where the surface area \( S \) varies with the square of the radius \( r \).

Identifying and understanding quadratic relationships is crucial because they appear in natural phenomena, such as the motion of objects under gravity and areas of geometric shapes. These relationships can be visualized graphically as parabolas, which open upwards when \( a > 0 \) and downwards when \( a < 0 \).

Quadratic Formulas in Geometry

In geometry, the quadratic formula links the dimensions of shapes to their areas or volumes. Recognizing a quadratic relationship allows for predictions and problem-solving, such as anticipating the growth pattern of the surface area of a sphere as the radius increases.

Mathematical Patterns

Mathematical patterns are sequences that follow a particular rule or a set of rules. In the exercise provided, we observe a pattern where the surface area of the sphere is directly proportional to the square of the radius. This pattern is a fundamental concept in mathematics, reflecting how a variable can affect another based on a consistent rule or relationship.

Identifying patterns, such as linear, quadratic, or exponential relationships, is essential in mathematics because they provide a foundational understanding that can be applied to solve complex problems more efficiently. By recognizing that the surface area of a sphere expands quadratically as its radius increases, we can predict the surface area for any given radius without having to physically measure the sphere each time.

Patterns in Nature and Technology

Patterns similar to those we see in the surface area of a sphere are ubiquitous in nature and technology, such as in the arrangement of seeds in a sunflower or the configuration of solar panels for optimal energy absorption. Understanding such patterns allows us to harness nature's efficiency and apply it to innovative designs and solutions.