Problem 22 - Technical Mathematics with Calculus

Question

Find the volume and the radius of a sphere having a surface area of 462.

Step-by-Step Solution

Answer

The radius of the sphere is approximately 6.08 units and the volume is approximately 935.8 cubic units.

Step 1: Understand the Relationship Between Surface Area and Radius

The surface area of a sphere can be calculated using the formula: \( A = 4\pi r^2 \), where \( A \) is the surface area and \( r \) is the radius of the sphere. We are given the surface area \( A = 462 \) square units.

Step 2: Solve for Radius

Substitute the given surface area into the formula and solve for \( r \): \[ 462 = 4\pi r^2 \] Divide both sides of the equation by \( 4\pi \): \[ \frac{462}{4\pi} = r^2 \] Take the square root of both sides to solve for \( r \): \[ r = \sqrt{\frac{462}{4\pi}} \]

Step 3: Calculate the Radius

Perform the calculation to find the radius: \[ r = \frac{\sqrt{462}}{2\sqrt{\pi}} \approx \frac{21.5}{3.54} \approx 6.08 \] So, the radius of the sphere is approximately 6.08 units.

Step 4: Use the Radius to Find Volume

The volume of a sphere is given by the formula: \( V = \frac{4}{3}\pi r^3 \). Now substitute the value of radius found in Step 3 into this formula to find the volume: \[ V = \frac{4}{3}\pi (6.08)^3 \]

Step 5: Calculate the Volume

Performing the calculation will determine the volume: \[ V \approx \frac{4}{3}\pi \times 224.4 \approx 297.5\pi \approx 935.8 \] Therefore, the volume of the sphere is approximately 935.8 cubic units.

Key Concepts

Sphere Calculations

Understanding sphere calculations is integral to mastering geometry concepts. In the context of a sphere — a perfectly symmetrical 3D object where every point on the surface is an equal distance from the center — calculations most often revolve around its surface area and volume.

To find the surface area of a sphere, we use the formula: \( A = 4\pi r^2 \) where \( A \) is the surface area and \( r \) is the radius. Knowing the surface area allows us to backtrack and find the sphere's radius by algebraically manipulating the formula. This is crucial since the radius is used in various other calculations, including determining the volume of the sphere.

The volume formula, \( V = \frac{4}{3}\pi r^3 \) , describes how much space the sphere occupies. Calculating the volume requires the radius, which we can derive from the surface area formula. It's a multi-step process that hinges on an understanding of the relationship between surface area, radius, and volume.

Geometry Problem Solving

Geometry problem solving is a systematic process that often involves breaking down a problem into manageable steps. When faced with a geometry problem, it's essential to understand the given information and the mathematical relationships between different shapes' properties, such as those of a sphere.

For instance, in the exercise we're discussing, we begin by identifying the known quantity (surface area) and translate it into the radius using the appropriate formula. This step allows us to make use of the given surface area — a critical piece of information — as we move towards finding the sphere's volume.

It's also pivotal to substitute numerical values correctly into formulas. Paying attention to units and ensuring consistent measurements can avoid errors. Geometry problem solving isn't just about finding a solution; it's about developing a deeper understanding of spatial relationships and how various geometric formulas are interconnected and applied.

Mathematical Formulas

Mathematical formulas are the tools that provide shortcuts to solving geometry problems. They are derived from the inherent properties of geometric figures and enable us to calculate lengths, areas, volumes, and other dimensions without having to start from the basics every time.

The formulas for both the surface area and volume of a sphere— \( A = 4\pi r^2 \) and \( V = \frac{4}{3}\pi r^3 \) — are prime examples of this. They represent general truths about all spheres, regardless of their size, making them powerful tools in our problem-solving arsenal.

In geometry, memorizing formulas can be helpful, but understanding how and why they work is even more advantageous. With this knowledge, you can solve a wider variety of problems and even derive these formulas yourself, should the need arise. Remember, formulas are derived from basic principles, so recognizing the relationships between different quantities can often lead you to the right formula for your calculation.