Problem 125 - 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 $$F=\frac{k M v^{4}}{r} \text { for } v$$

Step-by-Step Solution

Answer

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

Step 1: Isolate the Variable Term

The given formula is \(F = \frac{kM v^4}{r}\). To solve for \(v\), first isolate the term containing \(v\). Multiply both sides by \(r\) to get:\[ Fr = kMv^4 \]

Step 2: Solve for the Fourth Power of the Variable

With the equation \(Fr = kMv^4\), divide both sides by \(kM\) to solve for \(v^4\):\[ v^4 = \frac{Fr}{kM} \]

Step 3: Solve for the Variable

To solve for \(v\), take the fourth root of both sides of the equation. This results in:\[ v = \pm \sqrt[4]{\frac{Fr}{kM}} \]

Key Concepts

Solving Equations

Equations are mathematical statements that assert the equality of two expressions. Solving equations involves finding the value of the variable that makes the equation true. The process can differ based on the type of equations, such as linear, quadratic, or those involving exponents.

In our original exercise, the equation involves several terms: constants, variables, and rational exponents. Understanding each part of the equation helps simplify and solve it.

Here’s a general approach to solving equations:

• Identify the equation type and the operations involved, such as addition, subtraction, multiplication, division, or exponents.
• Perform inverse operations to simplify the equation.
• Check your solution by substituting back into the original equation.

By following these steps, you can solve even complex equations, such as those with rational exponents or multiple variables.

Isolating Variables

Isolating a variable means getting the variable by itself on one side of an equation. This process involves rearranging the equation so that you can solve for that specific variable. It’s a critical step in solving equations and is used to make the problem less complex.

In the given problem, we need to solve for the variable \(v\). The equation is \(F = \frac{kMv^4}{r}\). Here are the steps to isolate \(v\):

• First, eliminate fractions by multiplying both sides by the denominator, \(r\). This gives \(Fr = kMv^4\).
• Next, divide both sides by the coefficients \(kM\) to further isolate \(v\), resulting in \(v^4 = \frac{Fr}{kM}\).

Isolating the variable is a strategic move that simplifies the process of solving equations efficiently.

Rational Exponents

Rational exponents are exponents that are fractions. They represent both an integer exponent and a root. For example, \(a^{\frac{m}{n}}\) can be expressed as \(\sqrt[n]{a^m}\). Rational exponents are useful in simplifying expressions and solving equations.

In our problem, the rational exponent is \(v^4\), which needs to be addressed to solve for \(v\). Here’s how you deal with rational exponents:

• Understand that a rational exponent \(\frac{1}{n}\) is equivalent to taking the nth root.
• To eliminate a rational exponent, perform an operation (root) to both sides that will result in the variable being raised to the power of 1.

In our example, we take the fourth root of both sides since \(v^4 = \frac{Fr}{kM}\). This step results in \(v = \pm \sqrt[4]{\frac{Fr}{kM}}\).

Recognizing and manipulating rational exponents is crucial in algebra to simplify and resolve complex equations efficiently.