Cylinder Section Problem: Finding Distance To Plane

Hey guys! Today, we're diving into a fascinating geometry problem involving cylinders and planes. We've got a cylinder with a known volume and height, and a plane slicing through it parallel to the axis. The challenge? Figuring out the distance between the cylinder's axis and this cutting plane. Sounds interesting, right? Let's break it down step by step.

Understanding the Cylinder and the Cutting Plane

First, let's visualize the scene. We have a cylinder, think of it like a can of soda. We know its volume is $72\pi$ cubic centimeters and its height is 8 centimeters. This gives us some crucial information to work with. Now, imagine slicing this cylinder with a flat plane, but not just any slice – this plane is parallel to the central axis of the cylinder. This slice creates a cross-sectional area, a shape within the cylinder. The problem tells us that this cross-sectional area is half the size of the cylinder's axial section. What's an axial section, you ask? It's simply the rectangle you'd see if you sliced the cylinder straight down the middle, through its axis. Understanding these basic geometric relationships is key to solving the problem. We will use formulas for the volume of a cylinder and the area of rectangles and triangles. The axial section, as we mentioned, is the rectangle formed by a slice through the cylinder's axis. Its dimensions are the height of the cylinder and the diameter of its base. This is a critical concept to grasp, guys, as the area of this axial section will serve as a reference point for calculating the area of our cutting plane section. We are given that the cutting plane section's area is half of this axial section. This relative area will help us establish relationships between the radius of the cylinder, the distance of the plane from the axis, and the dimensions of the rectangular section formed by the cutting plane.

Key Concepts and Formulas

Before we jump into calculations, let's refresh some fundamental formulas that will be our best friends in this problem:

• Volume of a Cylinder: The volume (V) of a cylinder is given by $V = \pi r^2 h$, where 'r' is the radius of the base and 'h' is the height.
• Area of a Rectangle: The area (A) of a rectangle is simply length (l) times width (w), or $A = l \times w$.
• Area of a Triangle: We might need this later, so let's keep it in mind: $A = (1/2) \times base \times height$.
• Pythagorean Theorem: For right triangles, $a^2 + b^2 = c^2$, where 'a' and 'b' are the lengths of the legs and 'c' is the length of the hypotenuse.

Knowing these formulas is essential. We'll be using them to connect the given information (volume, height, area ratio) to the unknown – the distance from the axis to the plane. Remember, in geometry problems, it's all about finding the right relationships and applying the appropriate formulas. It's like having the right tools for the job!

Setting Up the Equations

Now, let's translate the problem's information into mathematical equations. This is where things get really interesting! We're given that the cylinder's volume is $72\pi$ cm³ and its height is 8 cm. Using the volume formula, we can write:

$72\pi = \pi r^2 (8)$

This equation allows us to solve for the radius (r) of the cylinder's base. Once we have the radius, we can determine the dimensions of the axial section, which is just a rectangle. The axial section's area will be the height of the cylinder (8 cm) multiplied by the diameter (2r). Remember, the diameter is twice the radius. We know the relationship between the axial section and the cutting plane's section: the cutting plane's area is half the axial section's area. Let's call the distance we're trying to find (the distance from the cylinder's axis to the cutting plane) 'd'. This 'd' will be a crucial component in our calculations involving the rectangular section formed by the cutting plane. The width of this rectangular section will depend on 'd' and the radius 'r'. We'll use the Pythagorean theorem to find this width, as the radius, the distance 'd', and half the width of the rectangle form a right triangle. So, we're essentially building a bridge between the cylinder's dimensions, the relative areas, and the distance 'd' using mathematical equations. This systematic approach is crucial for tackling complex geometry problems.

Solving for the Radius

Let's tackle the first part: solving for the radius 'r' using the volume equation we established earlier:

$72\pi = \pi r^2 (8)$

To isolate $r^2$, we can divide both sides of the equation by $8\pi$:

$r^2 = 72\pi / (8\pi)$

The $\pi$ terms cancel out, and 72 divided by 8 is 9:

$r^2 = 9$

Now, take the square root of both sides to find 'r':

$r = \sqrt{9} = 3$ cm

So, the radius of the cylinder's base is 3 cm. Great job, guys! We've taken the first step and unlocked a vital piece of information. With the radius in hand, we can now move on to calculating the area of the axial section and, subsequently, the area of the section formed by the cutting plane. Remember, finding the radius was essential because it links the volume information to the geometrical dimensions of the cylinder, allowing us to relate the areas and ultimately find the distance we're after.

Calculating the Areas

Now that we know the radius (r = 3 cm) and the height (h = 8 cm), let's calculate the area of the axial section. As we discussed earlier, the axial section is a rectangle with dimensions equal to the height of the cylinder and the diameter of the base. The diameter is simply twice the radius, so it's 2 * 3 cm = 6 cm. The area of the axial section (let's call it $A_{axial}$) is therefore:

$A_{axial} = height \times diameter = 8 cm \times 6 cm = 48 cm^2$

Next, we know that the area of the section formed by the cutting plane (let's call it $A_{section}$) is half the area of the axial section:

$A_{section} = (1/2) \times A_{axial} = (1/2) \times 48 cm^2 = 24 cm^2$

We've now successfully calculated both the axial section area and the cutting plane section area. This is a significant milestone, guys! We're getting closer to our final answer. These area values are crucial because they will help us determine the dimensions of the rectangular section formed by the cutting plane, which in turn will lead us to the distance we need to find.

Finding the Width of the Cutting Plane Section

The section formed by the cutting plane is also a rectangle. We know its area ($A_{section} = 24 cm^2$) and its height (which is the same as the cylinder's height, 8 cm). Let's call the width of this rectangle 'w'. We can use the area formula for a rectangle to find 'w':

$A_{section} = height \times width$

$24 cm^2 = 8 cm \times w$

Divide both sides by 8 cm to isolate 'w':

$w = 24 cm^2 / 8 cm = 3 cm$

So, the width of the rectangular section formed by the cutting plane is 3 cm. Awesome! We've found another key dimension. This width is essential because it connects the area of the section to its geometry within the cylinder, and this connection is how we'll ultimately calculate the distance from the axis.

Applying the Pythagorean Theorem

Now comes the clever part! Imagine drawing a line from the center of the cylinder's base to the midpoint of the width 'w' of the rectangular section. This line represents the distance we're trying to find (let's call it 'd'). Also, draw a line from the center of the base to one of the corners of the width 'w'. This line is simply the radius 'r' of the cylinder's base (3 cm). And finally, draw a line from the midpoint of 'w' to that same corner. This line is half the width, or w/2 = 3 cm / 2 = 1.5 cm. These three lines form a right triangle! The radius 'r' is the hypotenuse, the distance 'd' is one leg, and half the width (w/2) is the other leg. We can use the Pythagorean theorem to find 'd':

$r^2 = d^2 + (w/2)^2$

Substitute the values we know:

$3^2 = d^2 + (1.5)^2$

$9 = d^2 + 2.25$

Subtract 2.25 from both sides:

$d^2 = 9 - 2.25 = 6.75$

Take the square root of both sides:

$d = \sqrt{6.75} \approx 2.6 cm$

The Final Answer

Therefore, the distance from the cylinder axis to the cutting plane is approximately 2.6 cm. We did it, guys! We successfully solved the problem! Remember, the use of the Pythagorean theorem here is crucial. It allowed us to link the geometric dimensions we calculated (radius, width) to the distance we were trying to find, by leveraging the right triangle formed within the cylinder.

Key Takeaways

This problem was a fantastic exercise in spatial reasoning and applying geometric principles. Here are some key takeaways:

• Visualize the Problem: Always start by visualizing the geometric scenario. Draw a diagram if necessary. This helps you understand the relationships between different elements.
• Identify Key Information: Carefully identify the given information and what you need to find. In this case, we were given the volume, height, and a ratio of areas, and we needed to find a distance.
• Apply Relevant Formulas: Know your formulas! The volume of a cylinder, area of a rectangle, and the Pythagorean theorem were essential tools for this problem.
• Break Down the Problem: Complex problems can be broken down into smaller, manageable steps. We first found the radius, then the areas, then the width, and finally the distance.
• Look for Geometric Relationships: Geometry is all about relationships. Look for right triangles, similar shapes, and other geometric relationships that you can exploit.

Geometry problems can seem daunting at first, but with a systematic approach and a good understanding of the underlying principles, you can conquer them! Keep practicing, and you'll become a geometry master in no time. Great job working through this with me, guys!