Cylinder Radius & Generator: Solve Proportionality Problem

Hey guys! Today, we're diving into a classic geometry problem involving a right circular cylinder. It might sound intimidating, but we'll break it down step-by-step, making it super easy to understand. We're going to figure out how to calculate the radius and generator length when we know they're directly proportional and we have some extra information about the cylinder's diagonal section. So, grab your thinking caps, and let's get started!

Understanding the Problem Statement

So, the heart of our problem lies in understanding the relationships given. We are dealing with a right circular cylinder, which simply means a cylinder where the sides are perpendicular to the circular bases. Think of a can of soup – that's our cylinder! We're told that the radius (let's call it 'r') and the generator (let's call it 'g'), which is essentially the height of the cylinder, are directly proportional to the numbers 4 and 6 respectively. What this proportionality means is that the ratio between the radius and the generator will always remain the same, in this case, 4:6. This is a key concept to remember, as it allows us to set up a simple equation.

Adding to this, we also know that the diagonal of the cylinder's axial section is 30 cm. Now, imagine slicing the cylinder straight down the middle, through the center. The shape you see is a rectangle, and the diagonal they are talking about is the line that cuts across this rectangle, connecting opposite corners. This diagonal, along with the radius (which forms the base of the rectangle, but there are two radii, forming the full width which is the diameter, or 2r) and the generator (the height of the rectangle), forms a right-angled triangle. This is another critical piece of information that will allow us to use the Pythagorean theorem to solve the problem.

To summarise the given information and set the stage for our solution, we have these three essential pieces of data:

1. The radius (r) and generator (g) are directly proportional to 4 and 6.
2. The diagonal of the axial section is 30 cm.
3. The geometry of the cylinder allows us to form a right-angled triangle.

With a clear understanding of the problem at hand, we're now fully equipped to translate this information into mathematical equations and find our answers. Let’s dive into the solution process!

Setting Up the Equations

Alright, let's transform our understanding of the problem into mathematical equations. This is where things get really interesting! Remember, we have two crucial pieces of information that we need to express mathematically: the proportionality of the radius and generator, and the diagonal of the cylinder's section.

First, let's tackle the proportionality. We know that the radius r and the generator g are directly proportional to 4 and 6. This can be written as a ratio: r/g = 4/6. We can simplify this fraction by dividing both the numerator and denominator by 2, which gives us r/g = 2/3. This simplified ratio is much easier to work with. To further simplify our setup, we can introduce a constant of proportionality, let's call it k. This allows us to express both r and g in terms of k: r = 2k and g = 3k. This is a fantastic move because it reduces our two unknowns (r and g) into a single unknown (k), making the problem much easier to solve.

Now, let's bring in the information about the diagonal. We know that the diagonal of the axial section is 30 cm, and we've identified that this diagonal, along with the diameter (2r) and the generator (g), forms a right-angled triangle. This is where the Pythagorean theorem comes to our rescue! The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle, which in our case is the diagonal) is equal to the sum of the squares of the other two sides. Translating this into our cylinder's dimensions, we get the equation: (2r)^2 + g^2 = 30^2. This equation beautifully connects the radius, generator, and diagonal, giving us a powerful tool to solve for our unknowns.

So, now we have two key equations:

1. r = 2k and g = 3k (from the proportionality)
2. (2r)^2 + g^2 = 30^2 (from the Pythagorean theorem)

These equations are the foundation of our solution. We've successfully translated the geometrical relationships into algebraic expressions, and now we are ready to solve this system of equations to find the value of k, and subsequently, the values of r and g. Let's get into the nitty-gritty of solving the equations in the next section!

Solving the Equations

Okay, guys, we've set up our equations, and now it's time for the fun part: solving them! We have two equations and one unknown (k), which means we are in good shape to find our solution. Our equations are:

1. r = 2k and g = 3k
2. (2r)^2 + g^2 = 30^2

The strategy here is to use substitution. Since we have expressions for r and g in terms of k, we can substitute these expressions into the second equation. This will give us an equation with only k as the unknown, which we can then solve. Let’s start by substituting r = 2k and g = 3k into the Pythagorean equation:

(2 * (2k))^2 + (3k)^2 = 30^2

Now, let's simplify this equation step-by-step. First, we square the terms inside the parentheses:

(4k)^2 + (3k)^2 = 900

Next, we square the terms:

16k^2 + 9k^2 = 900

Now, combine the like terms:

25k^2 = 900

To isolate k^2, divide both sides of the equation by 25:

k^2 = 900 / 25 k^2 = 36

Now, take the square root of both sides to solve for k:

k = ±√36 k = ±6

Since k represents a proportionality constant related to lengths, we only consider the positive solution. So, k = 6. Now that we've found the value of k, we can easily find the values of r and g by substituting k = 6 back into our equations r = 2k and g = 3k:

r = 2 * 6 = 12 cm g = 3 * 6 = 18 cm

So, we've found that the radius r is 12 cm and the generator g is 18 cm. We have successfully solved our equations and found the dimensions of the cylinder! In the next section, we will recap the entire solution and make sure our answer makes sense in the context of the problem.

Summarizing the Solution

Alright, guys, let's take a step back and recap the entire journey we've been on to solve this cylinder problem. We started with a problem statement that gave us two crucial pieces of information: the proportionality between the radius and the generator, and the length of the diagonal of the cylinder's axial section. Our goal was to determine the radius and generator length.

Here’s a quick rundown of the steps we took:

1. Understanding the Problem: We carefully dissected the problem statement, making sure we understood the meaning of direct proportionality and how the cylinder's diagonal section formed a right-angled triangle.
2. Setting Up the Equations: We translated the given information into mathematical equations. We expressed the proportionality as r/g = 2/3 and then introduced a constant of proportionality, k, to get r = 2k and g = 3k. We then used the Pythagorean theorem on the right-angled triangle formed by the diameter, generator, and diagonal to get (2r)^2 + g^2 = 30^2.
3. Solving the Equations: We used substitution to solve the system of equations. We substituted r = 2k and g = 3k into the Pythagorean equation, simplified, and solved for k. We found that k = 6. Then, we plugged k = 6 back into the equations for r and g to find r = 12 cm and g = 18 cm.
4. The Answer: Therefore, the radius of the cylinder is 12 cm, and the generator length is 18 cm.

It's always a good idea to double-check our answer to make sure it makes sense. Let's plug our values of r and g back into the Pythagorean equation to see if they fit:

(2 * 12)^2 + 18^2 = 30^2

24^2 + 18^2 = 900

576 + 324 = 900

900 = 900

Our values check out! This confirms that our solution is correct. We successfully navigated through this geometry problem by breaking it down into smaller, manageable steps, translating the information into equations, and then solving those equations systematically. Remember, guys, this approach can be applied to many other math problems as well. It's all about understanding the problem, identifying the key relationships, and then using the right tools to find the solution. Keep practicing, and you'll become math-solving pros in no time!