Problem 26 - Technical Mathematics with Calculus

Question

Find the surface area and radius of a sphere that has a volume of \(462 \mathrm{ft}^{3}\).

Step-by-Step Solution

Answer

The radius of the sphere is approximately 4.79 ft, and the surface area is approximately 288.6768 ft^2.

Step 1: Set up the volume formula for a sphere

The volume of a sphere is given by the formula \[ V = \frac{4}{3}\pi r^3 \] where \(V\) is the volume and \(r\) is the radius of the sphere. To find the radius, we need to solve for \(r\) given that \(V = 462 \mathrm{ft}^3\).

Step 2: Solve for the radius

Substitute the given volume into the formula and solve for \(r\): \[ 462 = \frac{4}{3}\pi r^3 \] \[ r^3 = \frac{462 \times 3}{4\pi} \] \[ r^3 = \frac{1386}{4\pi} \] \[ r^3 = \frac{1386}{4\times3.14159} \] \[ r = \left( \frac{1386}{12.56636} \right)^{\frac{1}{3}} \] \[ r = \left( 110.275 \right)^{\frac{1}{3}} \] \[ r \approx 4.79 \mathrm{ft} \] (rounded to two decimal places)

Step 3: Set up the surface area formula for a sphere

The surface area of a sphere is given by the formula \[ A = 4\pi r^2 \] where \(A\) is the surface area and \(r\) is the radius of the sphere.

Step 4: Calculate the surface area

Using the radius found in Step 2, calculate the surface area: \[ A = 4\pi (4.79 \mathrm{ft})^2 \] \[ A = 4\pi (22.9441 \mathrm{ft}^2) \] \[ A = 4 \times 3.14159 \times 22.9441 \mathrm{ft}^2 \] \[ A \approx 288.6768 \mathrm{ft}^2 \] (rounded to four decimal places)

Key Concepts

Volume of a Sphere

When we talk about the volume of a sphere, we refer to the amount of space it occupies in three dimensions. The volume of a sphere can be calculated using a special formula: \[ V = \frac{4}{3}\pi r^3 \.\] In this equation, \(V\) represents the volume, \(\pi\) (approximately 3.14159) is the constant representing the ratio of a circle's circumference to its diameter, and \(r\) denotes the radius of the sphere, which is the distance from the center to any point on its surface.

Applying this formula requires precise computation, as the cube of the radius (\(r^3\)) and its multiplication with the constant \(\frac{4}{3}\pi\) can significantly influence the result. Usually, in mathematical problem solving, finding the volume leads to several other calculations or serves as a critical part of solving complex problems involving spheres.

Surface Area of a Sphere

The surface area is the total area covered by the surface of a three-dimensional object. For spheres, which are perfectly symmetrical, the surface area can be calculated using the formula: \[ A = 4\pi r^2 \.\] Here, \(A\) denotes the surface area and \(r\) is the sphere's radius. This formula takes the radius squared (\(r^2\)), multiplies it by four and \(\pi\), and yields the surface area in square units, reflecting how much material would be needed to cover the sphere or paint its surface.

Understanding the relationship between the radius and the surface area is crucial, as it shows that a small change in the radius can have a significant impact on the surface area due to the squaring of the radius in the formula. Whether designing a balloon or calculating the material for a spherical tent, mastery of this formula is essential.

Radius of a Sphere

The radius of a sphere is a fundamental concept in geometry. It is not only part of the formula used for calculating both the volume and the surface area of a sphere but is also critical for describing the sphere's size and understanding its geometric properties. To find the radius of a sphere, you may be given other measurements, such as the volume or surface area, and then use algebra to solve for the radius.

As in our original exercise, once the volume was known, we applied inverse operations to isolate the radius, taking the cube root of the adjusted value after equating the volume to its formula including the radius. This process exemplifies the interconnectedness of the sphere's properties and the essential role the radius plays in various calculations.

Mathematical Problem Solving

Mathematical problem solving involves understanding the problem, identifying the information provided, and deciding on a series of steps to obtain the desired solution. The process often requires the application of known formulas, algebraic manipulation, and logical reasoning based on mathematical principles.

For instance, in the given exercise, the problem was initially to find the radius and the surface area of a sphere given its volume. This entailed setting up the correct volume formula, solving for the unknown radius, and subsequently applying the radius to the surface area formula to find the final answer. This step-by-step strategy emphasizes organizing thoughts, performing calculations carefully, and verifying results for accuracy. It's a skill set that is invaluable across many facets of mathematics and can be extended to real-life problem-solving scenarios.