Problem 140 - College Algebra with Modeling and Visualization

Question

Approximate the constant of variation to the nearest hundredth. The variable \(z\) varies jointly as the 1.5 power of \(x\) and the 2.1 power of \(y .\) When \(x=4\) and \(y=3.5, z=397\).

Step-by-Step Solution

Answer

The constant of variation \(k\) is approximately 3.53.

Step 1: Identify the Formula

Since the variable \(z\) varies jointly as the 1.5 power of \(x\) and the 2.1 power of \(y\), we use the formula for joint variation: \( z = k \cdot x^{1.5} \cdot y^{2.1} \), where \(k\) is the constant of variation.

Step 2: Substitute Known Values

Given that \(x = 4\), \(y = 3.5\), and \(z = 397\), substitute these values into the joint variation formula: \( 397 = k \cdot 4^{1.5} \cdot 3.5^{2.1} \).

Step 3: Calculate the Value of Each Exponential Term

Calculate \(4^{1.5}\) and \(3.5^{2.1}\):- \(4^{1.5} = \sqrt{4^3} = \sqrt{64} = 8\)- \(3.5^{2.1} \approx 3.5^2 \cdot 3.5^{0.1} \approx 12.25 \cdot 1.1487 \approx 14.070175\)

Step 4: Solve for the Constant \(k\)

Substitute the calculated values back into the equation: \(397 = k \cdot 8 \cdot 14.070175\). Simplify the equation to solve for \(k\):\[397 = k \cdot 112.5614\]\(k = \frac{397}{112.5614} \approx 3.528\).

Step 5: Approximate to the Nearest Hundredth

Since we need to approximate \(k\) to the nearest hundredth, round \(3.528\) to \(3.53\).

Key Concepts

Constant of Variation

In mathematics, the concept of a "constant of variation" can simplify the representation of how variables relate to one another. It plays a crucial role in joint variation equations where one variable depends on multiple others. In joint variation, a variable, such as \( z \), is said to vary jointly as the products of other variables raised to certain powers. The constant of variation, denoted as \( k \), adjusts the scale of this relationship. It remains unchanged as long as the underlying relationship, described by the equation, doesn't change. In this problem's context, \( k \) is determined so that any change in \( x \) or \( y \) adjusts \( z \) proportionally, according to their specified powers.

Power of Variables

The power of variables is a concept that defines how a change in one variable affects another variable. In joint variation, variables can be raised to specific powers, also known as exponents. These powers determine the rate at which changes in an input variable influence the output variable. For example, if \( x \) is raised to the power of 1.5, it means that \( x \) has a non-linear, specific impact on \( z \). The greater the power, the more pronounced the effect of the variable. By calculating \( x^{1.5} \) and \( y^{2.1} \), we can observe how changes in \( x \) and \( y \) jointly impact the value of \( z \). The exercise reinforces the importance of understanding variable powers when determining relationships in equations.

Exponential Terms

Exponential terms are expressions where a number is repeatedly multiplied by itself, defined by the power or exponent. In the problem, you encountered exponential terms like \(4^{1.5}\) and \(3.5^{2.1}\). Calculating these terms involves applying exponent rules:

• To calculate \(4^{1.5}\), recognize this as \( \sqrt{4^3} \), which simplifies to 8.
• For \(3.5^{2.1}\), break it into simpler calculations: first \(3.5^2\), which is 12.25, then multiply by \(3.5^{0.1}\), approximately 1.1487, resulting in roughly 14.070175.

The complexity of exponential terms often requires using calculators for precise computation, especially with non-integer exponents. Mastery of exponential calculations strengthens problem-solving skills in various mathematical applications.