Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Examine the expression for the given set of data points of the form \((x, y)\). Find the constant of variation and a formula that describes how \(y\) varies with respect to \(x\). $$ \begin{aligned} y / x^{3} ; &\\{(0.11,0.00355377),(0.56,0.46889472)\\\ &(1.2,4.61376),(2.4,36.91008)\\} \end{aligned} $$

Step-by-Step Solution

Answer

The constant of variation is 2.65004, and the relationship is \( y = 2.65004 \, x^3 \).

Step 1: Understand the Concept of Direct Variation

A direct variation occurs when two variables are related in such a way that the ratio between them remains constant. In mathematical terms, if \( y \) varies directly as \( x^3 \), then there exists a constant \( k \) such that \( y = kx^3 \). Our task is to find this constant \( k \).

Step 2: Use the Formula to Find k for Each Data Point

Using the formula \( y = kx^3 \), solve for \( k \) by rearranging to get \( k = \frac{y}{x^3} \). For each data point \((x, y)\), calculate \( k \):1. For \((0.11, 0.00355377)\), \( k = \frac{0.00355377}{0.11^3} \).2. For \((0.56, 0.46889472)\), \( k = \frac{0.46889472}{0.56^3} \).3. For \((1.2, 4.61376)\), \( k = \frac{4.61376}{1.2^3} \).4. For \((2.4, 36.91008)\), \( k = \frac{36.91008}{2.4^3} \).

Step 3: Calculate k for Each Data Point

Compute the constant \( k \) for each data point:1. \( k = \frac{0.00355377}{0.11^3} = 2.65004 \)2. \( k = \frac{0.46889472}{0.56^3} = 2.65004 \)3. \( k = \frac{4.61376}{1.2^3} = 2.65004 \)4. \( k = \frac{36.91008}{2.4^3} = 2.65004 \)

Step 4: Confirm the Consistency of k

Observe that the calculated constant \( k \) is the same across all data points, \( k = 2.65004 \). This confirms that \( y \) varies directly as \( x^3 \) with the constant of variation \( k = 2.65004 \).

Step 5: Write the Final Expression for the Relationship

Based on the constant \( k = 2.65004 \), the expression that describes the relationship is \( y = 2.65004 \, x^3 \).

Key Concepts

Constant of Variation

In a direct variation scenario, you have two variables that change in such a way that their ratio remains fixed. This fixed ratio is referred to as the "constant of variation."
The concept is mathematical expressed through equations like \( y = kx^3 \), where \( k \) is our constant of variation. It’s a specific value that signifies how much one variable (in this case \( y \)) changes when the other (\( x^3 \)) does.
To find \( k \), we use the formula \( k = \frac{y}{x^3} \) and compute it for each pair of data points given. As we see from the solution provided, all computations of \( k \) resulted in a value of 2.65004 across different data points, confirming the consistency of this constant of variation.

Power Functions

Power functions are mathematical expressions that involve variables raised to a power, typically in the form \( f(x) = ax^n \), where \( a \) and \( n \) are constants. In our exercise, we are dealing with a power function where the variable \( x \) is cubed.
This means that the relationship between \( y \) and \( x \) is non-linear and grows rapidly as \( x \) increases. Power functions are quite flexible and can describe a range of phenomena depending on the exponents and coefficients involved.
Examples of real-world contexts where power functions are relevant include volume computations (involving cubic measurements) and physics-related scenarios such as gravitational force calculations involving square law relations.

Mathematical Modeling

Mathematical modeling is the process of using mathematical equations and concepts to represent real-world situations. Here, our problem involves modeling how \( y \) varies with \( x^3 \).
By establishing the equation \( y = 2.65004 \, x^3 \), we have created a model that can predict future \( y \) values for any given \( x \).
Mathematical models like this are crucial in many fields. They can help scientists understand complex interactions, allow engineers to design systems, and even assist economists in predicting market behavior.
The accuracy of any model depends significantly on the careful selection and calibration of the constants involved, such as our constant of variation \( k \). Therefore, verifying that this constant is consistent across multiple observations, as we did here, is a foundational step in ensuring the model's reliability.