Problem 95 - Algebra and Trigonometry

Question

Solve the equation for the indicated variable. \(V=\frac{4}{3} \pi r^{3} ; \quad\) for \(r\)

Step-by-Step Solution

Answer

\(r = \sqrt[3]{\frac{3}{4\pi} V} \).

Step 1: Multiply both sides by the reciprocal of the fraction

First, we need to isolate the term with \(r^3\) on one side of the equation. To do this, multiply both sides of the equation by the reciprocal of \(\frac{4}{3}\), which is \(\frac{3}{4}\). So, \( V = \frac{4}{3} \pi r^3 \) becomes \( \frac{3}{4}V = \pi r^3 \).

Step 2: Divide both sides by \(\pi\)

Next, to further isolate \(r^3\), divide both sides of the equation by \(\pi\). Now, the equation is \( \frac{3}{4\pi} V = r^3 \).

Step 3: Take the cube root of both sides

Finally, take the cube root of both sides to solve for \(r\). This results in \( r = \sqrt[3]{\frac{3}{4\pi} V} \).

Key Concepts

Formula Manipulation

Manipulating a formula is the process of rearranging it to express one variable in terms of others. This is essential when we want to solve for a particular variable, especially in equations that involve multiple terms. For example, with the formula for the volume of a sphere, \( V = \frac{4}{3} \pi r^3 \), we need to manipulate it to solve for \( r \). Here's how to approach this:

• Identify the variable you want to isolate.
• Perform operations such as multiplication, division, addition, or subtraction to both sides to rearrange the formula.
• Use inverse operations to simplify and get the desired variable on one side of the equation.

With careful manipulation, complex equations can be simplified, making them easier to understand and solve.

Isolating Variables

Isolating a variable means getting the variable you need by itself on one side of the equation. It simplifies the equation and makes it easier to solve for that specific variable. Let's look at how we isolate \( r \) from the original formula \( V = \frac{4}{3} \pi r^3 \):

• First, we align \( r^3 \) alone by eliminating fractions. Multiply both sides by the reciprocal of \( \frac{4}{3} \). As a result, \( \frac{3}{4}V = \pi r^3 \).
• Next, remove any coefficients attached to \( r^3 \) by dividing by \( \pi \). The new equation is \( \frac{3}{4\pi} V = r^3 \).
• Instead of guessing or checking, we use careful step-by-step isolation techniques to ensure accuracy and simplicity.

This systematic approach guarantees the variable is effectively and correctly isolated.

Cube Root

The cube root operation is the inverse of cubing a number. It helps find a number, which when multiplied by itself three times, gives the original number. If you are solving an equation like \( r^3 = \frac{3}{4\pi} V \), you will need to apply the cube root to both sides to solve for \( r \).

• Identify the cube power term, \( r^3 \), in the equation.
• Apply the cube root to both sides of the equation. This will transform \( r^3 \) to \( r \), giving you \( r = \sqrt[3]{\frac{3}{4\pi} V} \).
• Remember, applying the cube root is a handy method especially in physics or geometry, where volume calculations are frequent.

Understanding this concept ensures precise solutions for cubic equations.

Algebra

Algebra is a branch of mathematics that involves using symbols and letters to represent numbers and quantities in equations. It provides a way to reason through problems by using general rules. For the exercise equation \( V = \frac{4}{3} \pi r^3 \), algebraic principles guide the solution process:

• Equations often involve balancing both sides. Start by simplifying the equation using arithmetic operations.
• Using algebraic rules, transform the equation to isolate the variable of interest, \( r \) in this case.
• Solving algebraic equations involves logical steps that make problems more manageable and help us find solutions systematically.

Mastering algebra equips you with problem-solving skills applicable in multiple areas, not just math.