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Chapter 8: Problem 20

Solve each formula for the specified variable $$ F=\frac{k}{\sqrt{d}} \text { for } d $$

Short Answer

Expert verified
d = \left(\frac{k}{F}\right)^2

Step by step solution

01

Understand the formula

The given formula is \[ F = \frac{k}{\sqrt{d}} \] where we need to solve for the variable \( d \).
02

Isolate \( \sqrt{d} \)

Multiply both sides of the equation by \( \sqrt{d} \) to clear the denominator:\[ F \cdot \sqrt{d} = k \]
03

Solve for \( \sqrt{d} \)

Divide both sides of the equation by \( F \):\[ \sqrt{d} = \frac{k}{F} \]
04

Square both sides to solve for \( d \)

Square both sides of the equation to get rid of the square root:\[ d = \left(\frac{k}{F}\right)^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Equations
Algebraic equations are mathematical statements that show the equality of two expressions involving variables and constants. They are central to algebra and come in various forms like linear and quadratic equations.
In our example, the equation is \[ F = \frac{k}{\sqrt{d}} \] which shows a relationship between the variables \(F\), \(k\), and \(d\). Understanding how to manipulate these equations allows us to find the value of one variable in terms of the others. Here, we focus on solving for \(d\).
Isolating Variables
Isolating variables means rearranging the equation so that the variable you need to solve for stands alone on one side of the equation. This is an essential step to make the solving process straightforward.
In our problem, we start with the equation \[ F = \frac{k}{\sqrt{d}} \].
We need to solve for \(d\). To isolate \(\sqrt{d}\), we multiply both sides of the equation by \(\sqrt{d}\), getting:
\[ F \cdot \sqrt{d} = k \]
This helps us move closer to having \(d\) alone and ready for solving.
Solving for a Variable
Once we have isolated the variable, we can solve for it. In this case, after isolating \(\sqrt{d}\), we follow these steps:

  • Starting from \( F \cdot \sqrt{d} = k \), divide both sides by \(F\) to get:

  • \[ \sqrt{d} = \frac{k}{F}\]

This equation now shows \(\sqrt{d}\) in terms of \(k\) and \(F\). Next, we need to remove the square root to solve for \(d\) completely.
Square Roots
A square root is a value that, when multiplied by itself, gives the original number. The square root of \(d\), written as \(\sqrt{d}\), means that \(\sqrt{d} \cdot \sqrt{d} = d\).
In algebra, solving for a variable often involves dealing with square roots. Here, we have \[ \sqrt{d} = \frac{k}{F} \]
To solve for \(d\), we need to eliminate the square root by squaring both sides of the equation.
Squaring Both Sides
Squaring both sides of an equation eliminates the square root. This involves raising both sides of the equation to the power of 2.
Starting from \[ \sqrt{d} = \frac{k}{F} \]
we square both sides:
\[ d = \left(\frac{k}{F}\right)^2 \]
This step finalizes our solution, giving us \(d\) in terms of \(k\) and \(F\). This technique is broadly applicable in algebra to simplify and solve equations involving square roots.

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