Determining the Correct Expression

Oct 13, 2025 by windailys.com 17 views

Introduction

In mathematics, the volume of a cylinder is a fundamental concept that has numerous applications in various fields, including physics, engineering, and architecture. The formula for the volume of a cylinder is given by $v = \pi r^2 h$ , where $r$ is the radius of the cylinder and $h$ is the height. In this article, we will explore the concept of the volume of a cylinder and determine which expression represents the volume of a given can.

Understanding the Formula

The formula for the volume of a cylinder is $v = \pi r^2 h$ . This formula indicates that the volume of a cylinder is directly proportional to the square of its radius and its height. The constant of proportionality is $\pi$ , which is a mathematical constant approximately equal to 3.14.

The Given Can

Let's consider a can with a radius of $x$ and a height of $4$ . We are given two expressions, A and B, and we need to determine which one represents the volume of the can.

Expression A

Expression A is given by $3 \pi x^2 + 4 \pi x + 16 \pi$ . This expression appears to be a quadratic equation in terms of $x$ , with a coefficient of $3 \pi$ for the $x^2$ term, a coefficient of $4 \pi$ for the $x$ term, and a constant term of $16 \pi$ .

Expression B

Expression B is given by $3 \pi x^2 + 16 \pi$ . This expression is also a quadratic equation in terms of $x$ , with a coefficient of $3 \pi$ for the $x^2$ term and a constant term of $16 \pi$ .

Determining the Correct Expression

To determine which expression represents the volume of the can, we need to substitute the given values into the formula for the volume of a cylinder. The radius of the can is $x$ , and the height is $4$ . Therefore, the volume of the can is given by $v = \pi x^2 \cdot 4 = 4 \pi x^2$ .

Comparing the Expressions

Now, let's compare the two expressions with the formula for the volume of a cylinder. Expression A is $3 \pi x^2 + 4 \pi x + 16 \pi$ , while Expression B is $3 \pi x^2 + 16 \pi$ . The formula for the volume of a cylinder is $4 \pi x^2$ .

Conclusion

Based on the comparison, it is clear that Expression B, $3 \pi x^2 + 16 \pi$ , is the correct expression that represents the volume of the can. This is because it matches the formula for the volume of a cylinder, $4 \pi x^2$ , when the height is $4$ .

Final Thoughts

In conclusion, the volume of a cylinder is a fundamental concept in mathematics that has numerous applications in various fields. The formula for the volume of a cylinder is given by $v = \pi r^2 h$ , where $r$ is the radius of the cylinder and $h$ is the height. By understanding this formula and applying it to a given can, we can determine which expression represents the volume of the can. In this case, Expression B, $3 \pi x^2 + 16 \pi$ , is the correct expression that represents the volume of the can.

References

[1] "Mathematics for Engineers and Scientists" by Donald R. Hill
[2] "Calculus" by Michael Spivak
[3] "Geometry" by I.M. Gelfand

Additional Resources

Khan Academy: Calculus
MIT OpenCourseWare: Calculus
Wolfram Alpha: Calculus

Frequently Asked Questions

Q: What is the formula for the volume of a cylinder? A: The formula for the volume of a cylinder is $v = \pi r^2 h$ .
Q: How do I determine the volume of a cylinder? A: To determine the volume of a cylinder, you need to know the radius and height of the cylinder.
Q: What is the relationship between the radius and height of a cylinder? A: The volume of a cylinder is directly proportional to the square of its radius and its height.
The Volume of a Cylinder: A Q&A Article =====================================================

Introduction

In our previous article, we explored the concept of the volume of a cylinder and determined which expression represents the volume of a given can. In this article, we will continue to delve deeper into the world of cylinders and answer some frequently asked questions about the volume of a cylinder.

Q&A Session

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

A: The formula for the volume of a cylinder is $v = \pi r^2 h$ , where $r$ is the radius of the cylinder and $h$ is the height.

Q: How do I determine the volume of a cylinder?

A: To determine the volume of a cylinder, you need to know the radius and height of the cylinder. You can then use the formula $v = \pi r^2 h$ to calculate the volume.

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

A: The volume of a cylinder is directly proportional to the square of its radius and its height. This means that if you double the radius and height of a cylinder, its volume will increase by a factor of $2^3 = 8$ .

Q: Can I use the formula for the volume of a cylinder to find the volume of a sphere?

A: No, the formula for the volume of a cylinder is not applicable to spheres. The formula for the volume of a sphere is $V = \frac{4}{3} \pi r^3$ , where $r$ is the radius of the sphere.

Q: How do I calculate the volume of a cylinder with a non-circular base?

A: If the base of the cylinder is not a circle, you will need to use a different formula to calculate its volume. The formula for the volume of a cylinder with a non-circular base is $V = \pi r^2 h$ , where $r$ is the radius of the base and $h$ is the height.

Q: Can I use the formula for the volume of a cylinder to find the volume of a cone?

A: No, the formula for the volume of a cylinder is not applicable to cones. The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$ , where $r$ is the radius of the base and $h$ is the height.

Q: How do I calculate the volume of a cylinder with a non-uniform cross-sectional area?

A: If the cross-sectional area of the cylinder is not uniform, you will need to use a different formula to calculate its volume. The formula for the volume of a cylinder with a non-uniform cross-sectional area is $V = \int_{a}^{b} \pi (x^2) dx$ , where $a$ and $b$ are the limits of integration.

Q: Can I use the formula for the volume of a cylinder to find the volume of a torus?

A: No, the formula for the volume of a cylinder is not applicable to tori. The formula for the volume of a torus is $V = 2 \pi^2 R r^2$ , where $R$ is the major radius and $r$ is the minor radius.

Conclusion

In conclusion, the volume of a cylinder is a fundamental concept in mathematics that has numerous applications in various fields. By understanding the formula for the volume of a cylinder and answering some frequently asked questions, we can gain a deeper insight into the world of cylinders.

References

[1] "Mathematics for Engineers and Scientists" by Donald R. Hill
[2] "Calculus" by Michael Spivak
[3] "Geometry" by I.M. Gelfand

Additional Resources

Khan Academy: Calculus
MIT OpenCourseWare: Calculus
Wolfram Alpha: Calculus

Frequently Asked Questions

Q: What is the formula for the volume of a cylinder? A: The formula for the volume of a cylinder is $v = \pi r^2 h$ .
Q: How do I determine the volume of a cylinder? A: To determine the volume of a cylinder, you need to know the radius and height of the cylinder.
Q: What is the relationship between the radius and height of a cylinder? A: The volume of a cylinder is directly proportional to the square of its radius and its height.