Cylinder Within A Cylinder: Volume Ratio & Surface Area

Hey guys! Let's dive into some cool geometry problems. We're going to explore cylinders, prisms, and how their volumes and surface areas relate to each other. Get ready to flex those math muscles! We'll break down the problems step-by-step, making sure everything is clear and easy to follow. Let's get started!

2. Analyzing Cylinders and Prisms

Alright, imagine a cylinder perfectly snug inside a regular square prism, and then another smaller cylinder nestled right inside that. The big question is: How does the volume of the smaller cylinder compare to the volume of the larger cylinder? This is a classic geometry puzzle, and we can solve it by focusing on the relationships between their dimensions. Let's break it down to ensure we thoroughly understand how to approach this kind of problem. To make this easy, let's denote the side of the square base of the prism as 'a', the radius of the larger cylinder (the one circumscribed around the prism) as 'R', the radius of the smaller cylinder (inscribed within the prism) as 'r', and the height of the prism and both cylinders as 'h'.

First, consider the larger cylinder. Since it circumscribes the prism, its diameter is equal to the diagonal of the square base of the prism. The diagonal of a square with side 'a' is $a oot{2}$. Thus, the radius of the larger cylinder, R, is $a oot{2}/2$. The volume of a cylinder is given by the formula $V = \pi r^2 h$. Therefore, the volume of the larger cylinder, $V_1$, is $\pi R^2 h = \pi (a oot{2}/2)^2 h = \pi (2a^2/4)h = \pi (a^2/2)h$.

Next, let's look at the smaller cylinder. This cylinder is inscribed inside the prism, meaning its diameter is equal to the side of the square base of the prism, which is 'a'. Hence, the radius of the smaller cylinder, r, is a/2. The volume of the smaller cylinder, $V_2$, is $\pi r^2 h = \pi (a/2)^2 h = \pi (a^2/4)h$.

Now, to find the ratio of the volumes of these two cylinders, we simply divide the volume of the smaller cylinder by the volume of the larger cylinder: $V_2 / V_1 = [\pi (a^2/4)h] / [\pi (a^2/2)h]$. Notice that $π$, $a^2$ and $h$ all cancel out, leaving us with a ratio of (1/4) / (1/2), which simplifies to 1/2. Therefore, the ratio of the volumes of the two cylinders is 1:2. The volume of the smaller cylinder is half the volume of the larger cylinder. This result stems from the geometrical relationships between the square prism and the inscribed and circumscribed cylinders, specifically how the radii of the cylinders relate to the dimensions of the prism's base. It is a clear demonstration of how volume changes with changing dimensions and also exemplifies the practical usage of formulas when problem-solving. This ratio stays constant irrespective of the specific values of 'a' and 'h'.

Remember, in geometry, visualization is super helpful! Imagine the scenario: the smaller cylinder fills only half of the cross-sectional area of the larger cylinder, because the cross-sectional area of the larger cylinder spans the entire square prism, whereas the smaller cylinder's diameter is equal to only one side of the square base. The height, 'h,' is the same for both, so the volumetric difference comes from the differing base areas. Understanding this, along with being able to apply the volume formulas effectively, makes solving this type of problem a breeze.

In summary:

• The larger cylinder's radius is $a oot{2}/2$, volume: $\pi (a^2/2)h$.
• The smaller cylinder's radius is a/2, volume: $\pi (a^2/4)h$.
• The ratio of the volumes is 1:2.

Now, let's keep the momentum going!

3. Surface Area of a Cylinder Calculation

Let's switch gears and look at another intriguing problem. This time, we're given the lateral surface area of a cylinder, which equals $24

$. The goal is to figure out something about the cylinder, though the question is not completely specified, we might want to find the relationship between the radius and height if it is what the problem asks for. For the lateral surface area of the cylinder to be $24\pi$, we know that the formula is $2\pi rh$. where r is the radius and h is the height of the cylinder. Thus we have $2\pi rh = 24\pi$. To find the relationship between the radius and height, let us divide both sides of the equation by $2\pi$. This gives us $rh = 12$. This means the product of the radius and height of the cylinder is 12. There are many possible combinations of 'r' and 'h' that satisfy this condition, but the important thing is that their product always results in 12.

For example, if the radius (r) is 2, the height (h) would be 6, and if the radius is 3, the height would be 4, and vice versa. Without more information, such as the cylinder's volume or a relationship between the radius and the height, we can't determine the specific values of 'r' and 'h'. However, we do know that the product of the radius and height must equal 12. Let's look at it using different examples to give you the most possible scenarios in the problem.

• Scenario 1: If the cylinder has a radius of 1 unit, the height is 12 units. This results in the lateral surface area being $2\pi * 1 * 12 = 24\pi$.
• Scenario 2: If the cylinder has a radius of 2 units, the height is 6 units. The lateral surface area is given by $2\pi * 2 * 6 = 24\pi$.
• Scenario 3: If the cylinder has a radius of 3 units, the height is 4 units, so the lateral surface area is $2\pi * 3 * 4 = 24\pi$.
• Scenario 4: If the radius is 6, then the height is 2, with the lateral surface area being $2\pi * 6 * 2 = 24\pi$.

Each example maintains the lateral surface area of $24\pi$ regardless of individual radius and height values, so long as their product is 12. This emphasizes that there can be multiple solutions when only a relationship (in this case, the lateral surface area) is fixed, and the other parameters can change accordingly. You should remember the formulas and understand them to ensure that these types of problems are easy to tackle.

So, even though we can't pinpoint the exact values of the radius and height, we successfully found their relationship. This underscores a critical idea in geometry: sometimes, understanding a relationship between dimensions is enough to solve a problem, even without exact values. We leveraged the formula for the lateral surface area of the cylinder, $2\pi rh$, which allowed us to establish the equation $rh = 12$, thereby illustrating the flexibility and interconnectedness of geometric concepts.

In summary:

• The lateral surface area of a cylinder is given by $2\pi rh$.
• If $2\pi rh = 24\pi$, then $rh = 12$.
• The radius and height can vary, but their product must always equal 12.

And that's a wrap, guys! I hope you found these problems helpful. Keep practicing and exploring the amazing world of geometry! Have fun!