Problem 101 - College Algebra Essentials

Question

Solve each equation. $$ \text { Solve for } V: r=\sqrt{\frac{3 V}{\pi h}} $$

Step-by-Step Solution

Answer

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

Step 1: Square both sides of the equation

To get rid of the square root, square both sides of the equation. This results in the equation \( r^2 = \frac{3 V}{\pi h} \).

Step 2: Isolate the variable \( V \)

To get the \( V \) by itself, multiply both sides of the equation by \( \pi h \). This gives us: \( r^2 \pi h = 3V \).

Step 3: Final computation

Lastly, divide both sides of the equation by 3 to completely isolate \( V \). Thus, the solution is \( V = \frac{r^2 \pi h}{3} \).

Key Concepts

Isolation of Variables

Isolation of variables is a crucial step when solving equations, as it allows us to focus on the unknown part of the equation we want to find. In simple terms, isolating a variable means manipulating the equation so that the variable of interest is alone on one side of the equation.

To achieve isolation, perform operations such as addition, subtraction, multiplication, or division. The goal is to undo the operations that are applied to the variable to simplify the equation.

• Always perform the same operation on both sides of the equation to maintain equality.
• Organize your steps clearly to avoid confusion.

In our example, the variable we want to isolate is \( V \). Through algebraic manipulation, we moved other terms to the opposite side of the equation until \( V \) stood alone.

Algebraic Manipulation

Algebraic manipulation involves rearranging the given equation such that it's easier to manage and solve. It's essentially the toolkit we use to isolate the variables we care about.

Common techniques include:

• Adding or subtracting the same value from both sides.
• Multiplying or dividing each side by a constant.
• Using factorization to simplify expressions.

When solving our exercise, we began by squaring both sides to remove the square root, then multiplied by \( \pi h \), and finally divided by 3. Each step is an example of algebraic manipulation, making the problem manageable.

Square Roots

Square roots represent values that, when multiplied by themselves, give the original number. In equations, square roots can make them seem complex, but removing them is a straightforward process.

To eliminate a square root, follow these steps:

• Square both sides of the equation, transforming the equation into one without square roots.
• This step increases the power of each term by two, hence getting rid of the square root.

For example, in the equation \( r = \sqrt{\frac{3V}{\pi h}} \), squaring both sides resulted in \( r^2 = \frac{3V}{\pi h} \). This conversion simplifies the equation dramatically.

Pi Constant

The pi constant, represented by \( \pi \), is a fundamental element in mathematics, especially in fields involving circles and spheres. Its approximate value is 3.14159. Pi is an irrational number, meaning it has an endless, non-repeating decimal representation.

In our equations, \( \pi \) often appears in formulas involving areas, circumferences, and volumes where circles are involved. Since it is a constant, meaning its value is fixed, it helps in simplifying terms but is unaffected by algebraic operations aimed at isolating variables.

In our example, \( \pi \) remains part of the equation throughout the steps and aids in clearing the fraction. When we multiplied both sides by \( \pi h \), it was essential in isolating \( V \) by balancing both sides of the equation.