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Chapter 8: Problem 14

Solve each formula for the specified variable $$ L=\frac{k d^{4}}{h^{2}} \text { for } h $$

Short Answer

Expert verified
The solution for \(h\) is \(h = \sqrt{\frac{k d^{4}}{L}}\).

Step by step solution

01

- Isolate the variable

The variable to solve for is \(h\). First, let's multiply both sides of the equation by \(h^2\) to get rid of the fraction: \( L h^2 = k d^4 \)
02

- Solve for \(h^2\)

Next, divide both sides by \(L\) to isolate \(h^2\) on one side of the equation: \( h^2 = \frac{k d^{4}}{L} \)
03

- Solve for \(h\)

Finally, take the square root of both sides to solve for \(h\): \( h = \sqrt{\frac{k d^{4}}{L}} \)

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Key Concepts

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

Isolating Variables
In algebra, isolating a variable means to get the variable alone on one side of the equation.
If we isolate a variable correctly, we can find its value.
Here, we start with the equation: \[ L=\frac{k d^{4}}{h^{2}} \]
To isolate the variable 'h', we first need to remove the fraction by multiplying both sides by \( h^2 \).
This gives us: \( L h^2 = k d^4 \)
Make sure every term is properly balanced while manipulating the equation.
Solving for a Variable
Solving for a variable involves getting that variable by itself on one side of the equation.
After isolating \( h^2 \) by moving everything else to the other side, we get: \( h^2 = \frac{k d^4}{L} \)
This step allows us to focus solely on the variable we're solving for.
We need to make sure both sides of the equation are simplified.
Algebraic Manipulation
Algebraic manipulation refers to the use of basic algebraic operations to simplify or change equations.
This includes operations like addition, subtraction, multiplication, and division.
Here, we divided both sides of the equation by 'L' to isolate \(h^2\): \( h^2 = \frac{k d^4}{L} \)
Make sure that we perform the same operation on both sides to maintain the equality.
Square Root
Taking the square root is necessary when you have a variable squared and need to find its original value.
If we have: \( h^2 = \frac{k d^4}{L} \)
We solve for 'h' by taking the square root of both sides: \( h = \text{卤} \frac{\text{鈭殅(k d^4)}{\text{鈭殅(L)} \)
Remember, the square root symbol (鈭�) shows we need the principal square root.
Always consider both positive and negative solutions:
One side might be negative while the other is positive.

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