The Height of the Cylinder

Oct 13, 2025 by miodeya.com 101 views

Introduction

In mathematics, solving for the height of a cylinder is a common problem that involves using the formula for the volume of a cylinder. The formula for the volume of a cylinder is given by V = πr^2h, where V is the volume, π is a constant, r is the radius, and h is the height. In this article, we will walk through the steps to solve for the height of a cylinder using the given volume and radius.

The Problem

Louise calculated the height of a cylinder that has a volume of $$488 \pi$$ cubic centimeters and a radius of 9 centimeters. Her work is shown below.

${ V = \pi r^2 h }$

Step 1: Substitute the Given Values

The first step is to substitute the given values into the formula for the volume of a cylinder.

${ 488 \pi = \pi (9^2) h }$

In this step, we have substituted the given volume of $$488 \pi$$ cubic centimeters and the radius of 9 centimeters into the formula.

Step 2: Simplify the Equation

The next step is to simplify the equation by evaluating the expression inside the parentheses.

${ 488 \pi = \pi (81) h }$

In this step, we have simplified the equation by evaluating the expression inside the parentheses.

Step 3: Divide Both Sides by π

The next step is to divide both sides of the equation by π to isolate the term involving the height.

${ 488 = 81h }$

In this step, we have divided both sides of the equation by π to isolate the term involving the height.

Step 4: Divide Both Sides by 81

The final step is to divide both sides of the equation by 81 to solve for the height.

${ h = \frac{488}{81} }$

In this step, we have divided both sides of the equation by 81 to solve for the height.

Conclusion

In this article, we have walked through the steps to solve for the height of a cylinder using the given volume and radius. We have used the formula for the volume of a cylinder and substituted the given values into the formula. We have then simplified the equation and isolated the term involving the height. Finally, we have solved for the height by dividing both sides of the equation by 81.

The Final Answer

The final answer is:

${ h = \frac{488}{81} }$

The Height of the Cylinder

The height of the cylinder is $$\frac{488}{81}$$ centimeters.

Real-World Applications

Solving for the height of a cylinder has many real-world applications. For example, in architecture, engineers need to calculate the height of a cylinder to determine the volume of a building. In manufacturing, companies need to calculate the height of a cylinder to determine the volume of a container. In science, researchers need to calculate the height of a cylinder to determine the volume of a sample.

Tips and Tricks

Here are some tips and tricks to help you solve for the height of a cylinder:

Make sure to substitute the given values into the formula correctly.
Simplify the equation by evaluating the expressions inside the parentheses.
Isolate the term involving the height by dividing both sides of the equation by π.
Solve for the height by dividing both sides of the equation by the coefficient of the height term.

Common Mistakes

Here are some common mistakes to avoid when solving for the height of a cylinder:

Failing to substitute the given values into the formula correctly.
Failing to simplify the equation by evaluating the expressions inside the parentheses.
Failing to isolate the term involving the height by dividing both sides of the equation by π.
Failing to solve for the height by dividing both sides of the equation by the coefficient of the height term.

Conclusion

Q: What is the formula for the volume of a cylinder?

A: The formula for the volume of a cylinder is V = πr^2h, where V is the volume, π is a constant, r is the radius, and h is the height.

Q: How do I solve for the height of a cylinder?

A: To solve for the height of a cylinder, you need to substitute the given values into the formula for the volume of a cylinder, simplify the equation, isolate the term involving the height, and solve for the height.

Q: What are the steps to solve for the height of a cylinder?

A: The steps to solve for the height of a cylinder are:

Substitute the given values into the formula for the volume of a cylinder.
Simplify the equation by evaluating the expressions inside the parentheses.
Isolate the term involving the height by dividing both sides of the equation by π.
Solve for the height by dividing both sides of the equation by the coefficient of the height term.

Q: What is the most common mistake when solving for the height of a cylinder?

A: The most common mistake when solving for the height of a cylinder is failing to substitute the given values into the formula correctly.

Q: How do I avoid making mistakes when solving for the height of a cylinder?

A: To avoid making mistakes when solving for the height of a cylinder, make sure to:

Substitute the given values into the formula correctly.
Simplify the equation by evaluating the expressions inside the parentheses.
Isolate the term involving the height by dividing both sides of the equation by π.
Solve for the height by dividing both sides of the equation by the coefficient of the height term.

Q: What are some real-world applications of solving for the height of a cylinder?

A: Some real-world applications of solving for the height of a cylinder include:

Architecture: Engineers need to calculate the height of a cylinder to determine the volume of a building.
Manufacturing: Companies need to calculate the height of a cylinder to determine the volume of a container.
Science: Researchers need to calculate the height of a cylinder to determine the volume of a sample.

Q: How do I calculate the volume of a cylinder if I know the height and radius?

A: To calculate the volume of a cylinder if you know the height and radius, you can use the formula V = πr^2h, where V is the volume, π is a constant, r is the radius, and h is the height.

Q: What is the relationship between the volume and height of a cylinder?

A: The relationship between the volume and height of a cylinder is given by the formula V = πr^2h, where V is the volume, π is a constant, r is the radius, and h is the height.

Q: How do I calculate the radius of a cylinder if I know the volume and height?

A: To calculate the radius of a cylinder if you know the volume and height, you can use the formula r = √(V/πh), where r is the radius, V is the volume, π is a constant, and h is the height.

Q: What are some common formulas related to the volume of a cylinder?

A: Some common formulas related to the volume of a cylinder include:

V = πr^2h (formula for the volume of a cylinder)
r = √(V/πh) (formula for the radius of a cylinder)
h = V/πr^2 (formula for the height of a cylinder)

Conclusion

In conclusion, solving for the height of a cylinder is a common problem that involves using the formula for the volume of a cylinder. By following the steps outlined in this article, you can solve for the height of a cylinder using the given volume and radius. Remember to substitute the given values into the formula correctly, simplify the equation by evaluating the expressions inside the parentheses, isolate the term involving the height by dividing both sides of the equation by π, and solve for the height by dividing both sides of the equation by the coefficient of the height term.