Problem 121 - A Graphical Approach to Precalculus with Limits

Question

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$S=4 \pi r^{2} \quad \text { for } r$$

Step-by-Step Solution

Answer

\( r = \pm \sqrt{\frac{S}{4\pi}} \)

Step 1: Rearrange the Formula

The given formula is for the surface area of a sphere, which is \( S = 4 \pi r^2 \). We want to solve this formula for \( r \). Start by writing it as \( S = 4 \pi r^2 \).

Step 2: Isolate \( r^2 \)

To isolate \( r^2 \), divide both sides of the equation by \( 4\pi \): \( \frac{S}{4\pi} = r^2 \).

Step 3: Solve for \( r \)

Take the square root of both sides of the equation to solve for \( r \): \( r = \pm \sqrt{\frac{S}{4\pi}} \).

Key Concepts

Formulas

Formulas are essential mathematical tools that help us find relationships between different quantities. A formula is essentially an equation that expresses one variable in terms of others. In this context, formulas are crucial because they allow us to connect various physical properties, like how the surface area of a sphere depends on its radius. Knowing how to manipulate these formulas is key to solving many mathematical and real-world problems. When working with formulas:

• Identify the given formula and the variable you need to solve for.
• Understand that rearranging a formula means solving for a specific variable, which is what you'll need in many scientific and engineering situations.
• Use algebraic techniques to isolate the desired variable.

Mastering the art of rearranging formulas helps simplify complex problems and allows deeper insights into the relationships between different quantities.

Sphere Surface Area

The surface area of a sphere is a measure of the total area that the surface of the sphere occupies. Given by the formula, \( S = 4 \pi r^2 \), where \( S \) is the surface area and \( r \) is the radius of the sphere, it showcases a direct relation between these two quantities.

• "4" represents four times the square of the radius, emphasizing the spherical symmetry.
• The symbol \( \pi \) plays a fundamental role due to the sphere's circular nature.

Understanding how changing the radius affects the surface area can be critical, especially in fields involving physical space measurement, like architecture and materials science. A small increase in radius leads to a much larger increase in surface area due to the square in the formula, demonstrating nonlinear growth.

Isolate Variable

Isolating a variable means rearranging an equation to express one variable explicitly as a function of other variables. This is a common step when solving equations and can often involve:

• Addition, subtraction, multiplication, or division to both sides of an equation.
• Utilizing properties of equality to maintain balance throughout the operation.

In the context of the surface area formula \( S = 4 \pi r^2 \), we focused on isolating \( r^2 \) by dividing both sides by \( 4\pi \). This is a strategic move that allows us to next extract \( r \) by taking the square root. It's an important skill because it translates abstract formulas into practical solutions, aiding in gaining insights from data. The goal is to make computations easier and facilitate understanding.

Square Roots

Taking square roots is a vital mathematical operation that we frequently encounter when solving equations, especially when dealing with quadratic relationships or areas, like in this case of a sphere's surface area. To solve for the radius \( r \), we handle the equation \( r^2 \) by taking the square root of both sides, symbolized by \( \sqrt{} \).

• Remember, when you take the square root of a number, consider both the positive and negative roots, hence the \( \pm \) symbol.
• In a physical context, like measuring a sphere's radius, we often only consider the positive root, as a negative radius doesn't have a physical interpretation.

Understanding square roots not only aids in solving equations but also in fields like geometry and finance. It's about making numbers comprehensible and real.