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Chapter 8: Problem 12

Solve each formula for the specified variable $$ R=\frac{k}{d^{2}} \text { for } d $$

Short Answer

Expert verified
d = \( \sqrt{\frac{k}{R}} \)

Step by step solution

01

Understand the Formula

The given formula is \( R = \frac{k}{d^2} \). We need to solve for the variable \(d\).
02

Isolate the Denominator

Multiply both sides of the equation by \(d^2\) to get rid of the fraction: \( R \cdot d^2 = k \).
03

Solve for \(d^2\)

Divide both sides of the equation by \(R\) to isolate \(d^2\): \( d^2 = \frac{k}{R} \).
04

Take the Square Root

Take the square root of both sides to solve for \(d\): \( d = \sqrt{\frac{k}{R}} \).

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Key Concepts

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

algebraic manipulation
Algebraic manipulation involves rearranging and transforming equations using basic algebraic rules. In our exercise, we dealt with the formula: \( R = \frac{k}{d^2} \). To solve it, we performed arithmetic operations to move terms around and simplify the equation. This often includes operations like adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
These manipulations allow us to rearrange the equation methodically, step-by-step, until we isolate the desired variable.
isolating variables
Isolating a variable means getting the variable of interest by itself on one side of the equation, while all other terms are on the opposite side. To isolate the variable \(d\) in our formula \( R = \frac{k}{d^2} \), we need to eliminate the fraction.
  • First, we multiplied both sides by \(d^2\), resulting in \( R \, d^2 = k \).
  • We then divided both sides by \(R\) to further isolate \(d^2\), giving us \( d^2 = \frac{k}{R} \).
These steps show how isolating a variable can be an efficient process when followed systematically.
formula rearrangement
Formula rearrangement is closely related to algebraic manipulation and isolating variables. It is the process of reworking an equation to present it in a different form. This can often make solving for a particular variable more straightforward.
In the exercise, we rearranged the formula \( R = \frac{k}{d^2} \) into a new form to solve for \(d\).
  • Starting with multiplication, we moved from \( R = \frac{k}{d^2} \) to \( R \, d^2 = k \).
  • Next, we divided both sides to isolate \(d^2\) as \( d^2 = \frac{k}{R} \).

    • Continuing these methods systematically aids in making complex formulas manageable and solves for variables efficiently.
square roots
Square roots are used in situations where you need to find a number which, when multiplied by itself, gives you the original number. In the final step of our exercise, we dealt with taking the square root of both sides to solve for \(d\).
Given \( d^2 = \frac{k}{R} \), we applied the square root to both sides: \( d = \sqrt{\frac{k}{R}} \).
Here, square roots help us eliminate the exponent, reducing \( d^2 \) to \(d\). Understanding how and when to use square roots is essential for solving various algebraic equations, especially those involving squared terms.

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