Calculating The Radius Of A Combined Cylinder And Hemisphere Shape

In the fascinating world of geometry, we often encounter complex shapes formed by the combination of simpler ones. Understanding the surface area of these composite shapes requires a keen eye for detail and a firm grasp of fundamental formulas. In this article, we embark on a journey to dissect a shape formed by joining a cylinder and a hemisphere, unraveling the mystery of its radius given its surface area. This exploration will not only reinforce our understanding of geometric principles but also hone our problem-solving skills.

Before we dive into the specifics of our combined shape, let's refresh our understanding of the individual components: spheres, cylinders, and hemispheres. A sphere, a perfectly round three-dimensional object, boasts a surface area elegantly defined by the formula 4πr², where 'r' represents the radius. This formula encapsulates the total area encompassing the sphere's outer surface. Now, envision slicing the sphere precisely in half, and you'll arrive at a hemisphere. A hemisphere, therefore, possesses half the surface area of its parent sphere, which translates to 2πr². However, we must also account for the circular base exposed by the cut, which contributes an additional πr² to the total surface area. Thus, the curved surface area of a hemisphere is 2πr².

Cylinders, on the other hand, are characterized by their two circular bases and a curved lateral surface. To calculate the total surface area of a cylinder, we need to consider the areas of both circular bases, each contributing πr², and the lateral surface area. The lateral surface area can be visualized as a rectangle when the cylinder is "unrolled," with its length equivalent to the cylinder's circumference (2πr) and its width corresponding to the cylinder's height (h). Consequently, the lateral surface area is given by 2πrh. Summing these components, we arrive at the total surface area of a cylinder: 2πr² + 2πrh. This formula elegantly captures the total area encompassing the cylinder's outer surfaces.

Our composite shape is formed by seamlessly joining a cylinder and a hemisphere. This fusion creates a unique geometric entity with its own distinctive surface area characteristics. To determine the total surface area of this shape, we must carefully consider the individual surface areas of the cylinder and the hemisphere, while also accounting for any overlapping regions. The curved surface area of the hemisphere contributes 2πr² to the total surface area. For the cylinder, we need to consider its curved surface area, 2πrh, and the area of its base, πr². However, since one circular end of the cylinder is joined to the hemisphere, we only include one circular base in our surface area calculation.

When the cylinder and hemisphere are joined, the circular base of the hemisphere perfectly aligns with one of the circular ends of the cylinder. This creates an interface where these two surfaces effectively merge, and their areas are no longer exposed as part of the outer surface of the combined shape. Consequently, we must subtract the area of this overlapping circle, which is πr², from the combined surface area. Therefore, the total surface area of the combined shape is the sum of the curved surface area of the hemisphere (2πr²), the curved surface area of the cylinder (2πrh), and the area of one circular base of the cylinder (πr²), which simplifies to 3πr² + 2πrh.

Now, let's tackle the core of our problem: determining the radius of the cylinder given that the surface area of the combined shape is 135π m². We know that the surface area of the combined shape can be expressed as 3πr² + 2πrh. To solve for the radius, 'r', we need additional information, specifically the height of the cylinder, 'h'. Let's assume, for the sake of this example, that the height of the cylinder is equal to its radius, i.e., h = r. This assumption simplifies our equation and allows us to solve for 'r'.

Substituting h = r into our surface area equation, we get 3πr² + 2πr(r) = 135π. Simplifying this equation, we have 3πr² + 2πr² = 135π, which further reduces to 5πr² = 135π. Dividing both sides of the equation by 5π, we obtain r² = 27. Taking the square root of both sides, we find r = √27, which simplifies to r = 3√3. Therefore, the radius of the cylinder is 3√3 meters or approximately 5.20 meters (rounded to two decimal places).

Let's outline the step-by-step solution to solidify our understanding:

1. Identify the Components: Recognize that the shape is composed of a hemisphere and a cylinder.
2. Recall Surface Area Formulas: Remember the surface area formulas for a hemisphere (2πr²) and a cylinder (2πr² + 2πrh).
3. Account for Overlap: Understand that the circular base where the hemisphere and cylinder join is not part of the exposed surface area.
4. Formulate the Equation: Express the total surface area of the combined shape as the sum of the individual surface areas, subtracting the overlapping area: 3πr² + 2πrh = 135π.
5. Introduce an Assumption (if needed): If the height 'h' is unknown, assume a relationship between 'h' and 'r', such as h = r, to simplify the equation.
6. Substitute and Simplify: Substitute the assumed relationship (or the given value of 'h') into the equation and simplify.
7. Solve for 'r': Isolate 'r' and solve the resulting equation. This may involve taking square roots or other algebraic manipulations.
8. Express the Answer: State the value of 'r' in appropriate units (e.g., meters) and, if necessary, round to the specified decimal places.

This problem highlights the practical applications of geometric calculations. Understanding surface area is crucial in various fields, including architecture, engineering, and manufacturing. For instance, architects need to calculate the surface area of buildings to estimate the amount of materials required for construction. Engineers use surface area calculations to determine heat transfer rates in various systems. Manufacturers rely on surface area calculations to optimize packaging designs and minimize material usage. Moreover, surface area calculations are essential in fields like painting, coating, and insulation, where the amount of material needed is directly proportional to the surface area.

Determining the radius of a combined shape, such as a cylinder joined to a hemisphere, requires a solid understanding of geometric principles and meticulous application of surface area formulas. By carefully considering the individual components of the shape and accounting for any overlapping regions, we can formulate an equation that relates the radius to the total surface area. Solving this equation allows us to unveil the radius, a crucial parameter in various practical applications. This exercise not only reinforces our geometric knowledge but also enhances our problem-solving abilities, empowering us to tackle more complex challenges in the realm of geometry and beyond. The journey through surface area calculation is a testament to the power of mathematical reasoning in deciphering the world around us.

Keywords: Surface area, cylinder, hemisphere, radius, geometric calculation