Problem 116 - Elementary and Intermediate Algebra

Question

Communications Satellites. Engineers have determined that a spherical communications satellite needs to have a capacity of 565.2 cubic feet to house all of its operating systems. The volume \(V\) of a sphere is related to its radius \(r\) by the formula \(r=\sqrt[3]{\frac{3 V}{4 \pi}} .\) What radius must the satellite have to meet the engineer's specification? Use 3.14 as an approximation of \(\pi\).

Step-by-Step Solution

Answer

The required radius is approximately 5.13 feet.

Step 1: Identify the Key Formula

The formula relating the volume of a sphere to its radius is given by \( r = \sqrt[3]{\frac{3V}{4\pi}} \). This formula allows us to determine the radius when the volume \( V \) is known.

Step 2: Insert Given Values

Substitute the given volume of the satellite, 565.2 cubic feet, into the formula: \( r = \sqrt[3]{\frac{3 \times 565.2}{4 \times 3.14}} \).

Step 3: Calculate the Intermediate Value

Calculate the value inside the cube root. First, compute the numerator \( 3 \times 565.2 = 1695.6 \). Next, compute the denominator \( 4 \times 3.14 = 12.56 \). Then, divide the numerator by the denominator \( \frac{1695.6}{12.56} \approx 135 \).

Step 4: Compute the Cube Root

Take the cube root of the result from Step 3 to find the radius: \( r = \sqrt[3]{135} \). Using a calculator, this value is approximately \( r \approx 5.13 \).

Step 5: Confirm the Units

Ensure that the computed radius is in feet, as the volume was given in cubic feet.

Key Concepts

Spherical Coordinates

In geometry, spherical coordinates form a system that allows us to represent points in three-dimensional space. Unlike Cartesian coordinates, which use a grid-like system (x, y, z), spherical coordinates use three values: the radial distance, the polar angle, and the azimuthal angle. This system is particularly valuable when dealing with spheres or portions of spheres in physics and engineering.

• The radial distance (often denoted as \(r\)) is the distance from the origin to the point.
• The polar angle \(\theta\), measured from the positive z-axis, describes the inclination.
• The azimuthal angle \(\phi\), measured counterclockwise from the positive x-axis, defines the rotation around the z-axis.

Spherical coordinates are powerful in simplifying mathematical problems involving spherical symmetry, such as satellite systems or planetary movement. They offer an intuitive way to visualize and calculate measurements that involve angles and radiuses, making them indispensable in engineering calculations like finding volumes or surface areas of spheres, as seen in the communication satellite example.

Volume of a Sphere

The volume of a sphere is a fundamental concept in geometry, particularly when dealing with spherical objects or structures. The formula for the volume of a sphere is \[ V = \frac{4}{3} \pi r^3 \]where \( V \) represents the volume and \( r \) is the radius of the sphere. This formula is derived from integral calculus and provides the cubic measurement of the sphere, which is the space it occupies.

The importance of calculating the volume of a sphere arises in various fields, from architecture and design to physics and engineering. For example, when engineers design a communication satellite, knowing the volume helps in determining how much space is available for equipment.

To solve for the radius given the volume, as in the problem with the satellite, the volume formula is rearranged to express \( r \) in terms of known quantities. It simplifies to \[ r = \sqrt[3]{\frac{3V}{4\pi}} \]This formula allows us to find the radius when the volume is known, linking the spatial and dimensional characteristics of spheres smoothly.

Cube Root Calculation

The cube root is an essential mathematical operation when dealing with cubic measurements, such as volume. The cube root of a number \( x \) is another number \( y \) such that \( y^3 = x \). In the context of geometry and physical space, the cube root helps determine linear dimensions from cubic values, like finding the radius of a sphere when the volume is given.

Calculating the cube root can be done using calculators, which often have a dedicated function for this purpose, especially for non-perfect cubes. In some situations, approximation methods or numerical algorithms are employed for manual calculations.

• To approximate, you can estimate somewhere between values of known cubes, like between 5 \((5^3 = 125)\) and 6 \((6^3 = 216)\) for cube rooting 135.
• Spreadsheets and programming languages also offer built-in functions to compute cube roots efficiently.

Understanding cube root calculation is crucial in engineering and science as it aids in transitioning between different units and dimensions, as shown in determining the radius of a communication satellite from its known volume.