Problem 101 - Algebra and Trigonometry

Question

Solve for \(V: r=\sqrt{\frac{3 V}{\pi h}}\)

Step-by-Step Solution

Answer

The solution of the equation for \(V\) is \(V = \frac{\pi h r^2}{3}\).

Step 1: Square both sides

Squaring both sides of the equation gets rid of the square root. \((r)^2 = \frac{3 V}{\pi h}\). The squared equation turns into \(r^2 = \frac{3V}{\pi h}\).

Step 2: Rearrange the equation

Now, rearrange the equation to solve for \(V\). Multiply both sides by \(\pi h\) to isolate \(V\). The equation becomes \(V = \frac{\pi h r^2}{3}\).

Step 3: Simplification

The equation is already simplified, so no other operation needs to be carried out.

Key Concepts

Solving equations

In algebra, solving equations is all about finding the value of the unknown variable. For example, when dealing with a formula like \[ r = \sqrt{\frac{3V}{\pi h}} \]our goal is to isolate \( V \). This process involves manipulating both sides of the equation until \( V \) stands alone on one side. It’s like peeling back layers of an onion until you reach the core, or in this case, the solution.

• Start by understanding what operations are needed to isolate the variable, such as addition, subtraction, multiplication, or division.
• Always perform the same operation on both sides of the equation to maintain balance.
• Keep track of each step as you go, ensuring nothing is overlooked.

Each step we take brings us closer to solving for \( V \) by peeling away complexity and arriving at a clearer and simpler equation. The key is patience and practice!

Variable manipulation

Variable manipulation is a crucial skill in simplifying and solving equations. Here, we want to manipulate the variables so that we can solve for \( V \). After squaring both sides of the equation, we have:\[ r^2 = \frac{3V}{\pi h} \]
To manipulate the variables:

• Multiply both sides by \( \pi h \) to clear the fraction and bring \( V \) to one side.
• This results in \( \pi h r^2 = 3V \), making it easier to solve for our variable \( V \).
• Finally, divide by 3 to completely isolate \( V \).

Through these steps, variable manipulation helps to simplify the equation and make the solution more direct. It’s about using inverse operations and logical steps to reshape the equation into a solvable form.

Square root operations

Understanding square root operations is essential when dealing with equations involving roots. In the original problem, we see a square root:\[ r = \sqrt{\frac{3V}{\pi h}} \]
To remove the square root, we square both sides of the equation:

• This transforms the equation into \( r^2 = \frac{3V}{\pi h} \).
• Squaring is the opposite of square rooting; it cancels out the square root, simplifying the expression.

When working with square roots, remember:

• Squaring both sides is a useful technique to eliminate the root.
• Ensure every step is justified to avoid incorrect assumptions.
• After squaring, check your work by verifying the solution in the original equation if possible.

Mastering square root operations eases the process of solving more complex mathematical expressions by stripping them down to basic components.