Problem 16 - Beginning and Intermediate Algebra

Question

Solve each formula for the specified variable. (Leave \(\pm\) in the answers as needed.) See Examples I and 2. \(V=\pi r^{2} h\) for \(r\)

Step-by-Step Solution

Answer

r = \pm \sqrt{\frac{V}{\pi h}}

Step 1: Write down the given formula

Start with the given formula which states the volume of a cylinder: \[ V = \pi r^2 h \]

Step 2: Isolate the term containing the variable

To solve for \(r\), we need to isolate the term containing \(r\). Divide both sides by \(h\) to get: \[ \frac{V}{h} = \pi r^2 \]

Step 3: Isolate \(r^2\)

Next, divide both sides by \(\pi\) to isolate \(r^2\): \[ \frac{V}{\pi h} = r^2 \]

Step 4: Solve for \(r\)

Finally, take the square root of both sides to solve for \(r\): \[ r = \pm \sqrt{\frac{V}{\pi h}} \]

Key Concepts

Volume of a Cylinder

The volume of a cylinder is the amount of space that fits inside a cylinder. It can be visualized as how much water or material the cylinder can hold. The formula to calculate the volume is given by: The formula is: this here so relevant to understanding how these variables interact: The more familiar you are with well this formula, the easier it will be to solve problems involving cylinders.
Specifically, in the formula understand you how it is defined in this formula
both the horizontal distance across the shape. Each of these three components: height height) can be defined using the previously mentioned formula.

Isolating Variables

Isolating variables is a critical skill in algebra and calculus. It involves manipulating an equation to get one specific variable by itself on one side of the equation. This concept allows us to solve for unknown variables in formulas.
To isolate a variable, follow these steps:
Identify the variable you need to solve for.
Eliminate other terms or factors surrounding it using operations like addition, subtraction, multiplication, or division.
Simplify the equation if necessary to get the variable alone on one side.
In our exercise, we start with the formula for the volume of a cylinder: Simplify: on the this equation own side. it separate like to proceed other term Next get equation
next step is:
sides of final equation.

Square Root

Taking the square root is a fundamental operation in mathematics, especially when solving quadratic equations. The square root is a value that, when multiplied by itself, gives the original number. It is represented by the symbol \sqrt{...}.
In our problem, after isolating \(r^2\), to solve for \(r\) we take the square root: \( r = \pm \sqrt{\frac{V}{\pi h}} \).
The \(\pm\) (plus-minus) symbol indicates there are two possible values: a positive and a negative value.
Here's how to take square roots:
Identify the value or expression you need to find the square root of.
Use the square root symbol and place the value inside, simplifying if needed.
Note that any positive number has two square roots - one positive and one negative.
In our exercise, taking the square root is the final step to finding \(r\): The square step final