Which Expression Represents The Height Of The Cylinder Given The Expression 6 N 2 + 3 N + 4 6n^2 + 3n + 4 6 N 2 + 3 N + 4 ?A. 2 N − 7 − 48 3 N + 4 2n - 7 - \frac{48}{3n+4} 2 N − 7 − 3 N + 4 48 ​ B. 2 N − 7 2n - 7 2 N − 7 C. − 2 N + 7 − 5 3 N + 4 -2n + 7 - \frac{5}{3n+4} − 2 N + 7 − 3 N + 4 5 ​ D. − 2 N + 7 -2n + 7 − 2 N + 7

Introduction

In mathematics, solving for the height of a cylinder is a fundamental problem that involves understanding the relationship between the volume, radius, and height of a cylinder. Given the expression $6n^2 + 3n + 4$, we need to determine which expression represents the height of the cylinder. In this article, we will delve into the world of mathematics and explore the different options to find the correct solution.

Understanding the Problem

The problem requires us to find the height of a cylinder given the expression $6n^2 + 3n + 4$. To solve this problem, we need to understand the relationship between the volume, radius, and height of a cylinder. The volume of a cylinder is given by the formula $V = \pi r^2 h$, where $r$ is the radius and $h$ is the height.

Option A: $2n - 7 - \frac{48}{3n+4}$

Let's start by analyzing option A: $2n - 7 - \frac{48}{3n+4}$. To determine if this expression represents the height of the cylinder, we need to check if it satisfies the equation $V = \pi r^2 h$. Substituting the given expression for $h$, we get:

$$V = \pi r^2 (2n - 7 - \frac{48}{3n+4})$$

Expanding the expression, we get:

$$V = \pi r^2 (2n - 7) - \frac{48\pi r^2}{3n+4}$$

Comparing this expression with the formula for the volume of a cylinder, we can see that it does not satisfy the equation. Therefore, option A is not the correct solution.

Option B: $2n - 7$

Next, let's analyze option B: $2n - 7$. To determine if this expression represents the height of the cylinder, we need to check if it satisfies the equation $V = \pi r^2 h$. Substituting the given expression for $h$, we get:

$$V = \pi r^2 (2n - 7)$$

Expanding the expression, we get:

$$V = 2\pi r^2 n - 7\pi r^2$$

Comparing this expression with the formula for the volume of a cylinder, we can see that it does not satisfy the equation. Therefore, option B is not the correct solution.

Option C: $-2n + 7 - \frac{5}{3n+4}$

Now, let's analyze option C: $-2n + 7 - \frac{5}{3n+4}$. To determine if this expression represents the height of the cylinder, we need to check if it satisfies the equation $V = \pi r^2 h$. Substituting the given expression for $h$, we get:

$$V = \pi r^2 (-2n + 7 - \frac{5}{3n+4})$$

Expanding the expression, we get:

$$V = -2\pi r^2 n + 7\pi r^2 - \frac{5\pi r^2}{3n+4}$$

Comparing this expression with the formula for the volume of a cylinder, we can see that it does not satisfy the equation. Therefore, option C is not the correct solution.

Option D: $-2n + 7$

Finally, let's analyze option D: $-2n + 7$. To determine if this expression represents the height of the cylinder, we need to check if it satisfies the equation $V = \pi r^2 h$. Substituting the given expression for $h$, we get:

$$V = \pi r^2 (-2n + 7)$$

Expanding the expression, we get:

$$V = -2\pi r^2 n + 7\pi r^2$$

Comparing this expression with the formula for the volume of a cylinder, we can see that it satisfies the equation. Therefore, option D is the correct solution.

Conclusion

In conclusion, the correct expression that represents the height of the cylinder given the expression $6n^2 + 3n + 4$ is option D: $-2n + 7$. This expression satisfies the equation $V = \pi r^2 h$, which is the formula for the volume of a cylinder. Therefore, option D is the correct solution to the problem.

References

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

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 height of a cylinder? A: To determine the height of a cylinder, you need to substitute the given expression for $h$ into the formula for the volume of a cylinder and check if it satisfies the equation.
• Q: What is the correct expression that represents the height of the cylinder given the expression $6n^2 + 3n + 4$? A: The correct expression that represents the height of the cylinder given the expression $6n^2 + 3n + 4$ is option D: $-2n + 7$.
  Frequently Asked Questions: Solving for the Height of a Cylinder ====================================================================

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 and $h$ is the height.

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

A: To determine the height of a cylinder, you need to substitute the given expression for $h$ into the formula for the volume of a cylinder and check if it satisfies the equation. You can also use algebraic manipulations to isolate the variable $h$.

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

A: The volume of a cylinder is directly proportional to the square of the radius and the height. This means that if the radius or height of a cylinder increases, the volume will also increase.

Q: How do I choose the correct expression for the height of a cylinder?

A: To choose the correct expression for the height of a cylinder, you need to substitute the given expression for $h$ into the formula for the volume of a cylinder and check if it satisfies the equation. You can also use algebraic manipulations to isolate the variable $h$.

Q: What is the difference between the correct and incorrect expressions for the height of a cylinder?

A: The correct expression for the height of a cylinder is the one that satisfies the equation $V = \pi r^2 h$. The incorrect expressions do not satisfy this equation and therefore do not represent the height of the cylinder.

Q: Can I use other formulas to determine the height of a cylinder?

A: Yes, you can use other formulas to determine the height of a cylinder. For example, you can use the formula $V = \frac{4}{3}\pi r^3$ to find the height of a cylinder.

Q: How do I apply the formulas to real-world problems?

A: To apply the formulas to real-world problems, you need to substitute the given values into the formulas and solve for the unknown variable. You can also use algebraic manipulations to isolate the variable.

Q: What are some common mistakes to avoid when solving for the height of a cylinder?

A: Some common mistakes to avoid when solving for the height of a cylinder include:

• Not substituting the given expression for $h$ into the formula for the volume of a cylinder
• Not checking if the expression satisfies the equation $V = \pi r^2 h$
• Not using algebraic manipulations to isolate the variable $h$
• Not applying the formulas to real-world problems

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

A: To practice solving for the height of a cylinder, you can try the following:

• Use online resources such as Khan Academy or MIT OpenCourseWare to practice solving for the height of a cylinder
• Use algebraic manipulations to isolate the variable $h$
• Apply the formulas to real-world problems
• Check your work by substituting the given expression for $h$ into the formula for the volume of a cylinder and checking if it satisfies the equation

Q: What are some additional resources for learning about the height of a cylinder?

A: Some additional resources for learning about the height of a cylinder include:

• Khan Academy: Calculus
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Calculus
• "Mathematics for Engineers and Scientists" by Donald R. Hill
• "Calculus" by Michael Spivak
• "Geometry" by I.M. Yaglom

Conclusion

In conclusion, solving for the height of a cylinder is a fundamental problem in mathematics that involves understanding the relationship between the volume, radius, and height of a cylinder. By using algebraic manipulations and applying the formulas to real-world problems, you can determine the height of a cylinder and solve a variety of mathematical problems.