Problem 68 - Algebra and Trigonometry

Question

A verbal description of a function is given. Find (a) algebraic, (b) numerical, and (c) graphical representations for the function. Let \(V(d)\) be the volume of a sphere of diameter \(d\) . To find the volume, take the cube of the diameter, then multiply by \(\pi\) and divide by \(6 .\)

Step-by-Step Solution

Answer

Vol: \(V(d) = \frac{\pi}{6} d^3\); numerically evaluate for values (e.g., \(d=2,3,4\)); graph cubic curve.

Step 1: Formulate Algebraic Expression

Start by expressing the volume of the sphere as a function of the diameter. Given the description, the formula involves taking the cube of the diameter, multiplying by \( \pi \), and dividing by 6. This gives the algebraic expression: \[ V(d) = \frac{\pi}{6} d^3 \].

Step 2: Create Numerical Representation

Choose specific values for the diameter \(d\) and calculate the volume \(V(d)\) for each to provide a numerical representation. For example, if \(d = 2\), then \[ V(2) = \frac{\pi}{6} \times 2^3 = \frac{\pi}{6} \times 8 = \frac{4\pi}{3} \]. Calculate for several values like \(d = 3, 4, 5\) etc., to get a table of values.

Step 3: Sketch Graphical Representation

Use the algebraic formula \(V(d) = \frac{\pi}{6} d^3\) to sketch the graph. Since the relationship is a cubic function, plot several points calculated from the numerical representation on a graph with \(d\) on the x-axis and \(V(d)\) on the y-axis. Connect the points smoothly to visualize the increase in volume with diameter.

Key Concepts

Algebraic Representation

In the realm of mathematics, an algebraic representation is used to define functions through equations. In this case, we're looking at the function that describes the volume of a sphere based on its diameter. The problem explained that the volume should be computed by taking the cube of the diameter, multiplying it by \( \pi \), and dividing everything by 6. This leads to the algebraic formula: \[ V(d) = \frac{\pi}{6} d^3 \] This formula is quite powerful because it allows us to calculate the volume for any given diameter \(d\). Essentially, algebraic representations translate verbal descriptions into mathematical expressions, providing a clear and precise method to work with functions.

Numerical Representation

Numerical representation involves selecting specific values to gain a better understanding of a function's behavior. By picking distinct values for the diameter \(d\), we can compute the corresponding volumes \(V(d)\) and observe how the volume changes with different diameters. For example, let's say \(d = 2\), then the volume is: \[ V(2) = \frac{\pi}{6} \times 2^3 = \frac{4\pi}{3} \] By calculating further, if \(d = 3\), \[ V(3) = \frac{\pi}{6} \times 3^3 = \frac{9\pi}{2} \] These calculations can be extended for \(d = 4, 5\) and so on. Each calculation offers a specific, tangible volume for a given diameter, thereby creating a numerical mapping of the function. Creating a table of these values can further aid in visualizing and understanding how the function behaves.

Graphical Representation

A graphical representation of a function is pivotal for understanding its overall behavior and trends. To represent the volume function graphically, use the algebraic expression \( V(d) = \frac{\pi}{6} d^3 \). Start by plotting points on a graph using the numerical examples calculated earlier. - Place the diameter \(d\) on the x-axis - Plot the corresponding volume \(V(d)\) on the y-axis Since the function is cubic, its graph will appear as a smoothly increasing curve, reflecting that as the diameter grows, the volume increases exponentially. Graphs like this help in visualizing rapid changes in volumes as diameters increase, which can be less intuitive with sheer numerical data. Drawing connections between these points will give you a comprehensive picture of the function's behavior.