Problem 51 - College Algebra

Question

Solve for \(r : V=\pi r^{2} h\)

Step-by-Step Solution

Answer

The solution is \( r = \sqrt{\frac{V}{\pi h}} \).

Step 1: Identify the given equation

We start with the given formula for the volume of a cylinder, which is \[ V = \pi r^2 h \] where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.

Step 2: Isolate the term with the variable to solve for

We need to solve for \( r \). Begin by isolating the term \( \pi r^2 \) on one side of the equation by dividing both sides by \( h \):\[ \frac{V}{h} = \pi r^2 \]

Step 3: Remove the coefficient of the squared term

To solve for \( r^2 \), divide both sides of the equation by \( \pi \):\[ \frac{V}{\pi h} = r^2 \]

Step 4: Solve for the variable

Take the square root of both sides to solve for \( r \):\[ r = \sqrt{\frac{V}{\pi h}} \] This gives the formula for \( r \) in terms of \( V \) and \( h \).

Key Concepts

Solving Equations

Solving equations is like finding the missing piece of a puzzle. It's about figuring out what number or value makes an equation true. In our exercise, the equation given is the volume formula of a cylinder: \( V = \pi r^2 h \). This equation relates the volume \( V \), radius \( r \), and height \( h \) of a cylinder. Our task is to solve for \( r \), which means we need to find an expression for \( r \) in terms of \( V \) and \( h \).

To do this, we work backwards. Start with the full equation and begin simplifying or reducing it step-by-step until \( r \) is isolated on one side of the equation. We're essentially asking, "what manipulation or transformation will get \( r \) by itself?" This is at the heart of solving equations: applying inverse operations to both sides of the equation to maintain balance, even as we simplify it. Keep in mind, each step must be mathematically valid. Whatever we do to one side of the equation, we must do to the other.

Once we reach a point where only our needed variable \( r \) is left on one side of the equation, we are done. This solved equation can help us find the radius if we know the volume and height.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations or expressions to simplify or solve them. In our problem, the key is manipulating the volume formula \( V = \pi r^2 h \) to solve for the radius \( r \). Begin by identifying the term we need to isolate, which is \( r \). This means the rest of the terms \( V \), \( \pi \), and \( h \) must be handled accordingly.

First, we divide both sides of the equation by \( h \): \( \frac{V}{h} = \pi r^2 \). This isolates the term \( \pi r^2 \) on one side. Notice how the careful choice of division helped in reducing the number of terms on one side of the equation.

Next, divide both sides by \( \pi \) to isolate \( r^2 \): \( \frac{V}{\pi h} = r^2 \). Each step of manipulation brings us closer to the desired variable, reducing unnecessary complexity in the equation.

Finally, take the square root of both sides \( r = \sqrt{\frac{V}{\pi h}} \) to solve for \( r \). Remember, the square root operation is the inverse of the square operation, and this final step gives the exact value of the radius, providing us the key to understand the cylinder's size.

Geometry

Geometry is a branch of mathematics dealing with shapes, sizes, and the properties of space. Our exercise revolves around a cylinder, a 3D geometric shape. The cylinder's key components are its circular base and its height. Visualizing a cylinder can help you understand why its volume is calculated using the formula \( V = \pi r^2 h \).

Think of the volume as the amount of space "filled" inside the cylinder. It's like stacking circular disks on top of one another up to the height \( h \). Each disk has an area \( \pi r^2 \), which comes from the formula for the area of a circle \( A = \pi r^2 \), where \( r \) is the radius of the base circle.

The height \( h \) tells us how tall the cylinder is—that's how many of those circular disks you'll be stacking. By multiplying the base area by the height \( h \), you account for the entire "filled" space within the cylinder. That's why the complete formula for the volume of a cylinder is \( V = \pi r^2 h \). Understanding these geometric principles helps in visualizing and comprehending how altering \( r \) or \( h \) changes the cylinder's volume.