Cylinder Base Area: A Geometry Problem Solved

Hey everyone! Let's dive into a cool geometry problem. We're going to figure out the area of a cylinder's base, and I'll walk you through it step-by-step. Buckle up, because it's going to be a fun ride. The problem gives us some crucial information about a cylinder, and we'll use that to unlock the solution. This is a classic example of how understanding geometric relationships and applying some trigonometry can help us solve seemingly complex problems. So, grab your pencils and let's get started!

Understanding the Problem

Alright, first things first, let's break down what we're dealing with. The problem tells us that we have a cylinder. Cylinders, as you know, have two circular bases, and a curved surface connecting them. The centers of these bases are labeled as $O_1$ and $O_2$. We're given some specific measurements and an angle: $AO_2 = 12$ cm, and $\angle O_1 O_2 A = \arcsin \frac{1}{2}$. Our mission? To find the area of the cylinder's base. The area of a circle, which is what our base is, is given by the formula $A = \pi r^2$, where 'r' is the radius. So, our main goal is to find the radius of the base. Knowing the radius will give us everything we need to calculate the area. This question uses the geometric features of a cylinder, including its base and the angles formed by its parts. This is a typical geometry problem, so we have to use everything we know from school to solve the problem.

Now, let's look at the given values. $AO_2$ is a line segment, and its length is 12 cm. $\angle O_1 O_2 A = \arcsin \frac{1}{2}$ tells us that the angle at point $O_2$ is such that its sine is 1/2. We can use this to find the angle measure itself. Remember, the arcsin function is the inverse sine function. It gives us the angle whose sine is a given value. For this, we'll need a bit of trigonometry knowledge. The angle whose sine is 1/2 is 30 degrees (or $\frac{\pi}{6}$ radians). We'll also use other mathematical formulas to get our answer. The core of this problem lies in linking these components with the right mathematical rules to solve it. It is essential to use a step-by-step approach. This will help us avoid any confusion. That's the key to solving this.

Visualizing the Cylinder

Before we jump into the math, it helps to visualize what's going on. Imagine the cylinder standing upright. The line segment $AO_2$ connects a point on the circumference of the top base (let's call it point A) to the center of the bottom base ($O_2$). The line segment $O_1 O_2$ is the height of the cylinder, connecting the centers of the two bases. The angle $\angle O_1 O_2 A$ is formed at point $O_2$. With this visualization in mind, we can see that we have a right triangle, where the height of the cylinder is one side, $AO_2$ is the hypotenuse, and $AO_1$ is the radius of the base. This is important because it sets up the mathematical relationships that we'll use to solve the problem. Drawing a diagram can be a massive help. It allows us to visualize the problem and can often reveal relationships between different components that we might have missed otherwise. When in doubt, sketch it out. The geometry of a cylinder is pretty straightforward, but a diagram can help clarify how the different parts relate to each other. Don't underestimate the power of a good sketch! It will help you organize all the details of the problem.

Finding the Radius

Now, let's get to the heart of the matter: finding the radius. We know the length of $AO_2$ (12 cm) and the angle $\angle O_1 O_2 A$ (30 degrees). Consider the right triangle $O_1 O_2 A$. The radius of the cylinder's base is the line segment $O_1 A$. Since we know the hypotenuse ($AO_2 = 12$ cm) and the angle at $O_2$, we can use trigonometric functions to find the radius. The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse. In our case, the side opposite to the 30-degree angle is the radius ($O_1 A$). So, we can write:

$\sin(\angle O_1 O_2 A) = \frac{O_1 A}{AO_2}$

We know that $\sin(30^\circ) = \frac{1}{2}$, and $AO_2 = 12$ cm. Plugging these values into the equation, we get:

$\frac{1}{2} = \frac{O_1 A}{12}$

Now, solve for $O_1 A$ (the radius):

$O_1 A = 12 \cdot \frac{1}{2} = 6$ cm.

So, the radius of the cylinder's base is 6 cm. We're almost there! We have found the radius of the base of the cylinder, which means we can move on to the final step: finding the area of the base. Always double-check your calculations and units to ensure that everything is in order. We've got the necessary information to calculate our final answer. With the radius in our hands, all that's left is to find the area.

Calculating the Area

We've found the radius, which is 6 cm. The formula for the area of a circle is $A = \pi r^2$. Plugging in the radius, we get:

$A = \pi \cdot (6 \text{ cm})^2$

$A = \pi \cdot 36 \text{ cm}^2$

$A = 36\pi \text{ cm}^2$

Therefore, the area of the cylinder's base is $36\pi$ square centimeters. And there you have it, folks! We've successfully solved the problem. By applying basic trigonometric principles and understanding the geometry of a cylinder, we were able to find the area of the base. Remember to always use the right formulas, draw diagrams if you need to, and take your time to understand the relationships in the problem. With practice, you'll become a geometry whiz in no time. If you're going to solve geometry problems, keep the formulas close.

The Answer

So, the correct answer is (c) $36\pi$ cm². We went through the problem step by step, which made the solution pretty easy. We knew how to relate different geometric aspects to each other, and that helped a lot. These types of problems might look tricky at first glance, but with the right approach, they become a lot more manageable. This is a common type of question. You'll see questions similar to this if you are a student, or if you are interested in Geometry.

Final Thoughts

Geometry can be a blast once you get the hang of it. Remember to practice regularly, review the basic formulas and concepts, and don't be afraid to ask for help when you need it. There are tons of resources out there – textbooks, online tutorials, and practice problems – to help you along the way. Keep at it, and you'll find that geometry can be both challenging and incredibly rewarding. Keep practicing and keep learning, and you'll find that geometry is a really fun subject. Now go forth and conquer those geometry problems!