Problem 51 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Solve for \(r: V=\pi r^{2} h\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Formula

The formula given is for the volume of a cylinder, where \( V \) is the volume, \( \pi \) is a constant (approximately 3.14159), \( r \) is the radius of the base, and \( h \) is the height. We need to solve for \( r \).

Step 2: Isolate the Term with r

To solve for \( r \), we first isolate the term that includes \( r^2 \). We do this by dividing both sides of the equation by \( \pi h \).\[ \frac{V}{\pi h} = r^2 \]

Step 3: Solve for r

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

Key Concepts

Understanding the Cylinder Volume Formula

The volume of a cylinder is a measure of how much space is inside it. This is calculated using the formula \( V = \pi r^{2} h \). Here, \( V \) stands for volume, \( \pi \approx 3.14159 \) is a mathematical constant, \( r \) represents the radius of the cylinder's base, and \( h \) is the height of the cylinder.

To visualize it more simply, think of a cylinder as a tall soda can. The base is the circular top, and the height is the length from top to bottom. The formula essentially calculates how many small unit cubes can fit into this space. The term \( \pi r^{2} \) calculates the area of this circular base, and then multiplying it by the height \( h \) gives the total volume.

Radius Calculation from Volume

When you're tasked with finding the radius \( r \) from the volume \( V \), you'll need to rearrange the volume formula. The given exercise is to find out \( r \) when \( V = \pi r^2 h \).

We start by isolating the \( r^2 \) term to one side of the equation. To do so, we divide both sides by \( \pi h \), resulting in:

• \( \frac{V}{\pi h} = r^2 \)

Next, to get \( r \) on its own, take the square root of both sides, which will give:

• \( r = \sqrt{\frac{V}{\pi h}} \)

This formula tells us how to calculate the radius if we know the volume and the height of the cylinder.

Step-by-Step Solution

Breaking down problems into manageable steps helps in understanding and solving them. Let’s outline the steps taken for isolating \( r \) in the cylinder volume formula:

• Step 1: Understanding the Formula
  Start by identifying what each part of the formula represents. \( V \) is volume, \( \pi \approx 3.14159 \), \( r \) is the radius, and \( h \) is the height of the cylinder.
• Step 2: Isolating \( r^2 \)
  We aim to express \( r \) in terms of known quantities. Dividing both sides by \( \pi h \) isolates \( r^2 \).
• Step 3: Solving for \( r \)
  The final step involves taking the square root of both sides, giving \( r = \sqrt{\frac{V}{\pi h}} \).

By following these steps, you systematically find \( r \) from the volume and height, making potentially complex problems much simpler to handle.