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Chapter 8: Problem 16

Solve each formula for the specified variable $$ V=\pi r^{2} h \text { for } r $$

Short Answer

Expert verified
r = \( \sqrt{\frac{V}{\pi h}} \)

Step by step solution

01

- Understand the formula

The given formula is the volume of a cylinder: \[\begin{equation}V = \pi r^{2} h\end{equation}\]where \(V\) is the volume, \(r\) is the radius, and \(h\) is the height. The goal is to solve for \(r\).
02

- Isolate \(r^2\)

To isolate \(r^2\), first divide both sides of the equation by \(\pi h\):\[\begin{equation}\frac{V}{\pi h} = r^{2}\end{equation}\]
03

- Solve for \(r\)

Finally, take the square root of both sides to isolate \(r\):\[\begin{equation}r = \sqrt{\frac{V}{\pi h}}\end{equation}\]Note that we consider the positive square root because radius is a positive measurement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cylinder Volume
To start, let's understand the concept of the volume of a cylinder. A cylinder is a 3-dimensional shape with two parallel circular bases and a curved surface connecting them. The volume of a cylinder is the amount of space it occupies, calculated using the formula: \
\[ V = \pi r^{2} h\]
Where:
  • \( V \) stands for the volume
  • \( \pi \) is a constant (approx. 3.14159)
  • \( r \) is the radius of the cylinder's base
  • \( h\) is the height of the cylinder
This formula multiplies the area of the circular base (\( \pi r^{2} \)) by the height \(h\). The result is the total volume within the cylinder's boundaries.
In this exercise, your task is to solve for the radius \(r\), when given the volume \(V\) and height \(h\). Let's break down how to do that!
Isolating Variable
When solving equations, isolating the variable of interest is key. Here, we need to isolate \(r\) from the cylinder volume formula.
Let's start with the original equation: \[ V = \pi r^{2} h \]
Our goal is to isolate \(r^{2}\) first. To do this, divide both sides of the equation by \pi and \(h\). This gives: \[ \frac{V}{\pi h} = r^{2} \]
By dividing by \pi and \(h\), we effectively remove both terms from the right side, leaving \(r^{2}\) alone. Now, we are one step closer to finding \(r\).
Square Root
After isolating \(r^{2}\), the last step is to solve for \(r\). We can do this by taking the square root of both sides of the equation: \[ r = \sqrt{\frac{V}{\pi h}} \]
This step is crucial because the square root function reverses the squaring of the radius. Note that we only take the positive square root, as the radius \(r\) must be a positive dimension.
And there you have it! The radius \(r\) of the cylinder has been isolated and solved using the given volume \(V\) and height \(h\). Understanding these steps will help you tackle any similar problem involving isolating a variable or using the square root function.

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Most popular questions from this chapter

A rectangle has a length \(2 \mathrm{~m}\) less than twice its width. When \(5 \mathrm{~m}\) are added to the width, the resulting figure is a square with an area of \(144 \mathrm{~m}^{2}\). Find the dimensions of the original rectangle.

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