Which Of The Following Equations Is Equivalent To $S=\pi R^2 H$?A. $h=S-\pi R^2$B. $h=\frac{S}{\pi R^2}$C. \$h=\frac{\pi R^2}{S}$[/tex\]D. $h=S+\pi R^2$

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Solving for Height in a Cylinder: Which Equation is Equivalent to the Volume Formula?

When working with the volume of a cylinder, we often come across the formula $S=\pi r^2 h$, where SS represents the volume, rr is the radius, and hh is the height. However, in certain situations, we may need to solve for the height hh in terms of the volume SS and the radius rr. In this article, we will explore which of the given equations is equivalent to the original volume formula.

Understanding the Original Formula

The original formula $S=\pi r^2 h$ represents the volume of a cylinder. To find the height hh, we need to isolate it on one side of the equation. This can be done by dividing both sides of the equation by πr2\pi r^2. However, before we proceed, let's examine the given options and see which one matches this process.

Analyzing the Options

Option A: $h=S-\pi r^2$

This equation suggests that the height hh is equal to the volume SS minus the area of the base πr2\pi r^2. However, this is not equivalent to the original formula. If we subtract πr2\pi r^2 from both sides of the original equation, we would be left with S−πr2=πr2h−πr2S - \pi r^2 = \pi r^2 h - \pi r^2, which simplifies to S−πr2=πr2(h−1)S - \pi r^2 = \pi r^2 (h - 1). This is not the same as the original formula.

Option B: $h=\frac{S}{\pi r^2}$

This equation suggests that the height hh is equal to the volume SS divided by the area of the base πr2\pi r^2. This is equivalent to the process of dividing both sides of the original equation by πr2\pi r^2, which isolates the height hh on one side of the equation. Therefore, this option is a strong candidate for being equivalent to the original formula.

Option C: $h=\frac{\pi r^2}{S}$

This equation suggests that the height hh is equal to the area of the base πr2\pi r^2 divided by the volume SS. However, this is not equivalent to the original formula. If we divide both sides of the original equation by SS, we would be left with SS=πr2hS\frac{S}{S} = \frac{\pi r^2 h}{S}, which simplifies to 1=πr2hS1 = \frac{\pi r^2 h}{S}. This is not the same as the original formula.

Option D: $h=S+\pi r^2$

This equation suggests that the height hh is equal to the volume SS plus the area of the base πr2\pi r^2. However, this is not equivalent to the original formula. If we add πr2\pi r^2 to both sides of the original equation, we would be left with S+πr2=πr2h+πr2S + \pi r^2 = \pi r^2 h + \pi r^2, which simplifies to S+πr2=πr2(h+1)S + \pi r^2 = \pi r^2 (h + 1). This is not the same as the original formula.

Based on our analysis, we can conclude that the correct answer is Option B: $h=\frac{S}{\pi r^2}$. This equation is equivalent to the original volume formula $S=\pi r^2 h$, as it isolates the height hh on one side of the equation by dividing both sides by πr2\pi r^2. Therefore, if we need to solve for the height hh in terms of the volume SS and the radius rr, we can use this equation.

The concept of solving for height in a cylinder has numerous real-world applications. For instance, in architecture, engineers need to calculate the height of a cylindrical building or a water tank based on its volume and radius. Similarly, in manufacturing, companies need to determine the height of a cylindrical container or a pipe based on its volume and radius. By using the correct equation, we can ensure that our calculations are accurate and reliable.

When working with equations, it's essential to follow the order of operations (PEMDAS) to ensure that our calculations are correct. Additionally, we should always check our work by plugging in sample values or using a calculator to verify our results. By following these tips and tricks, we can ensure that our calculations are accurate and reliable.

When solving for height in a cylinder, one common mistake is to confuse the area of the base with the volume of the cylinder. This can lead to incorrect calculations and results. To avoid this mistake, we should always be careful when working with equations and ensure that we are using the correct formulas and variables.

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

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

Q: How do I solve for height in a cylinder?

A: To solve for height in a cylinder, you can use the equation $h=\frac{S}{\pi r^2}$, where SS is the volume, rr is the radius, and hh is the height.

Q: What is the difference between the area of the base and the volume of a cylinder?

A: The area of the base of a cylinder is πr2\pi r^2, while the volume of a cylinder is πr2h\pi r^2 h. The area of the base is a two-dimensional measurement, while the volume is a three-dimensional measurement.

Q: Can I use the equation $h=S+\pi r^2$ to solve for height in a cylinder?

A: No, you cannot use the equation $h=S+\pi r^2$ to solve for height in a cylinder. This equation is not equivalent to the original volume formula $S=\pi r^2 h$, and it will not give you the correct result.

Q: What is the significance of the radius in the equation $h=\frac{S}{\pi r^2}$?

A: The radius is an important variable in the equation $h=\frac{S}{\pi r^2}$ because it affects the area of the base of the cylinder. The larger the radius, the larger the area of the base, and the smaller the height of the cylinder.

Q: Can I use the equation $h=\frac{\pi r^2}{S}$ to solve for height in a cylinder?

A: No, you cannot use the equation $h=\frac{\pi r^2}{S}$ to solve for height in a cylinder. This equation is not equivalent to the original volume formula $S=\pi r^2 h$, and it will not give you the correct result.

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

A: The volume of a cylinder is directly proportional to its height. As the height of the cylinder increases, the volume of the cylinder also increases.

Q: Can I use the equation $h=S-\pi r^2$ to solve for height in a cylinder?

A: No, you cannot use the equation $h=S-\pi r^2$ to solve for height in a cylinder. This equation is not equivalent to the original volume formula $S=\pi r^2 h$, and it will not give you the correct result.

Q: What is the significance of the volume in the equation $h=\frac{S}{\pi r^2}$?

A: The volume is an important variable in the equation $h=\frac{S}{\pi r^2}$ because it affects the height of the cylinder. The larger the volume, the larger the height of the cylinder.

In conclusion, solving for height in a cylinder is a straightforward process that involves using the correct equation. By using the equation $h=\frac{S}{\pi r^2}$, you can ensure that your calculations are accurate and reliable. Remember to always check your work and use the correct formulas and variables to avoid common mistakes.