Question

The formula for the volume of a cylinder is \(V=\pi r^2 h\), where \(r\) is the radius of the base and \(h\) is the height of the cylinder. Which formula could be used to find \(h\) if you're given \(V\) and \(r\) ? (A) \(h=\frac{\pi r^2}{V}\) (B) \(h=V \pi r^2\) (C) \(h=\frac{V}{\pi r^2}\) (D) \(h=\frac{r^2}{\pi V}\)

Short answer

The correct formula to find \(h\) if given \(V\) and \(r\) is: \(h = \frac{V}{\pi r^2}\) (Answer: C).

Step-by-step solution

Write down the given formula for the volume of a cylinder

The formula for the volume of a cylinder is given by: \(V = \pi r^2 h\).

Isolate h in the formula

To isolate \(h\) in the formula, we need to divide both sides of the equation by \(\pi r^2\). Doing so, we get: \(h = \frac{V}{\pi r^2}\)

Match the obtained formula with the given options

Now, we need to compare the formula we obtained for \(h\), which is \(\frac{V}{\pi r^2}\), with the given options. We can see that it matches with option (C). Therefore, the correct formula to find \(h\) if given \(V\) and \(r\) is: \(h = \frac{V}{\pi r^2}\) (Answer: C).

Key concepts

Understanding the Geometry Formula for Cylinder Volume

To find the volume of a cylinder, we use the geometry formula: \( V = \pi r^2 h \). This equation is fundamental in geometry and is connected to the shape and dimensions of a cylinder.

Here's a quick breakdown of what each variable represents:

• \( V \) is the volume of the cylinder.
• \( \pi \) is a constant, approximately 3.14159.
• \( r \) is the radius of the base of the cylinder.
• \( h \) is the height of the cylinder.

In this formula, \( \pi r^2 \) essentially calculates the area of the cylinder's base—a circle. By multiplying this area by the height \( h \), we extend this area throughout the height of the cylinder to calculate the volume. Understanding this formula helps in solving many practical problems involving cylinders in mathematics and real-world scenarios.

How to Solve Equations to Isolate Variables

When we solve equations, sometimes we need to manipulate them to find a specific unknown. In the exercise, we started with the equation for cylinder volume: \( V = \pi r^2 h \). Our goal was to solve this equation to find the height \( h \) instead.

Here's how isolation of the variable \( h \) is achieved:

• Identify what you know: We have \( V \) and \( r \), but need \( h \).
• Reorganize the equation: Divide both sides by \( \pi r^2 \) to solve for \( h \).
• Resulting equation: \( h = \frac{V}{\pi r^2} \).

This method of isolating a variable by performing the same operation to both sides of an equation is a core algebraic technique, key in solving many types of mathematical problems. It ensures we maintain the balance of the equation while transforming it to reveal the desired unknown variable.

Importance of Mathematics Education

Mathematics education is vital as it equips students with the skills to apply mathematical concepts to solve real-life problems. Learning how to manipulate equations is just one of the many fundamental skills acquired through a structured mathematics education.

Here are a few reasons why mathematics education matters:

• Develops critical thinking and analytical skills, which are applicable across all academic disciplines.
• Prepares students for STEM (Science, Technology, Engineering, Mathematics) careers, which are integral to innovation and economic growth.
• Improves problem-solving abilities, helping students tackle challenges in everyday life beyond the classroom.

The exercise on the cylinder's volume and solving for \( h \) is a great example demonstrating how mathematical principles are ingrained in everyday objects and industries. By fostering a strong mathematics education, students gain valuable insights and skills that go beyond the confines of textbooks.