Problem 123 - A Graphical Approach to College Algebra

Question

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$F=\frac{k M v^{4}}{r} \text { for } v$$

Step-by-Step Solution

Answer

\( v = \pm \sqrt[4]{\frac{Fr}{kM}} \)

Step 1: Understand the Given Formula

The formula given is \( F = \frac{k M v^4}{r} \). We need to solve this equation for the variable \( v \).

Step 2: Isolate the Term Involving v

To isolate the term involving \( v^4 \), multiply both sides of the equation by \( r \) to cancel the denominator. This gives \( Fr = kMv^4 \).

Step 3: Solve for v^4

Divide both sides of the equation by \( kM \) to isolate \( v^4 \) on one side. This results in \( v^4 = \frac{Fr}{kM} \).

Step 4: Solve for v

Take the fourth root of both sides of the equation to solve for \( v \). This gives us \( v = \pm \sqrt[4]{\frac{Fr}{kM}} \).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is the art of rearranging equations to make them easier to work with. In this exercise, algebraic manipulation helps transform the original formula into a simpler form to isolate the desired variable. First, observe the given equation: \( F = \frac{k M v^4}{r} \). The goal is to express the formula in terms of \( v \).

To begin this process, multiply both sides by \( r \) to remove the fraction. This step is crucial because it allows you to simplify the equation by eliminating the denominator:

• Original: \( F = \frac{k M v^4}{r} \)
• After multiplying by \( r \): \( F r = k M v^4 \)

With this manipulation, the formula is now easier to manage as it no longer includes a fraction. You can continue by dividing both sides by \( k M \) to achieve \( v^4 = \frac{F r}{k M} \). Each of these steps changes the arrangement of the equation without altering its balance, showcasing the power of algebraic manipulation.

Equation Solving

Equation solving involves finding the value of variables that make an equation true. In our case, we are tasked with solving the equation for \( v \). Once we reach the expression \( v^4 = \frac{F r}{k M} \), the core of equation solving begins. This part requires us to perform operations on both sides of the equation to find \( v \):

• Recognize the operation, which in this case involves raising to the power of four.
• To reverse this, take the fourth root of both sides.

After applying these operations, the equation \( v = \pm \sqrt[4]{ \frac{F r}{k M} } \) is obtained. The use of \( \pm \) in the solution is essential when taking even roots because both positive and negative results satisfy the original equation. This demonstrates the fundamental practice in solving equations: if you carefully reverse operations, you can find the variable of interest.

Isolation of Variables

Isolation of variables is a key technique in algebra, aiming to rearrange an equation such that one variable stands alone on one side of the equality sign. This is exactly what we did to solve for \( v \) in the given formula.The strategy includes:

• Removing fractions by multiplying both sides by the denominator to simplify the equation.
• Interchanging operations to gradually isolate the power variable \( v^4 \).
• Finally, extracting the root to solve for \( v \) itself.

At each stage, the equation is expertly manipulated to peel away layers surrounding the variable until \( v \) is isolated.

The process illustrates a step-by-step removal of obstacles preventing the variable from standing alone:

• First make room by multiplication.
• Then distribute by division.
• Finally, unlock by taking a root.

This technique of isolating variables is fundamental in algebra and serves as an essential tool for solving a wide array of mathematical problems.