Problem 142 - College Algebra with Modeling and Visualization

Question

The variable \(z\) varies jointly as the third powers of \(x\) and y. If \(z=2160\) when \(x=3\) and \(y=4,\) find \(z\) when \(x=2\) and \(y=5\).

Step-by-Step Solution

Answer

The value of \( z \) is 1250 when \( x = 2 \) and \( y = 5 \).

Step 1: Understanding Joint Variation

When a variable varies jointly with other variables, it means that the variable is proportional to the product of those variables raised to certain powers. In this exercise, we are told that the variable \( z \) varies jointly as the third powers of \( x \) and \( y \). This can be expressed by the equation \( z = kx^3y^3 \), where \( k \) is the constant of variation.

Step 2: Finding the Constant of Variation

To find the constant \( k \), we can use the given values: \( z = 2160 \), \( x = 3 \), and \( y = 4 \). Substitute these into the joint variation formula: \( 2160 = k \cdot 3^3 \cdot 4^3 \). Calculate \( 3^3 = 27 \) and \( 4^3 = 64 \), so the equation becomes \( 2160 = k \cdot 27 \cdot 64 \). Solve for \( k \) by dividing both sides by \( 27 \cdot 64 \): \( k = \frac{2160}{27 \cdot 64} \).

Step 3: Calculating the Constant

Calculate \( 27 \cdot 64 = 1728 \). Thus, \( k = \frac{2160}{1728} = 1.25 \). So, the constant of variation \( k \) is 1.25.

Step 4: Using the Constant to Find New Value of z

Now, we need to find \( z \) when \( x = 2 \) and \( y = 5 \). Substitute \( k = 1.25 \), \( x = 2 \), and \( y = 5 \) into the joint variation equation: \( z = 1.25 \cdot 2^3 \cdot 5^3 \). Calculate \( 2^3 = 8 \) and \( 5^3 = 125 \), leading to \( z = 1.25 \cdot 8 \cdot 125 \).

Step 5: Calculating the Final Value of z

Determine \( 8 \cdot 125 = 1000 \), so \( z = 1.25 \cdot 1000 = 1250 \). Therefore, when \( x = 2 \) and \( y = 5 \), the value of \( z \) is 1250.

Key Concepts

Constant of Variation

In mathematics, when we talk about the constant of variation, we are describing a fixed number that defines how two or more variables are related in a proportional relationship, specifically in joint or direct variation. This constant remains the same no matter what values the variables take, as long as the relationship holds.
In joint variation, a variable is proportional to the product of two or more other variables. For example, when the problem states that the variable \( z \) is proportional to the third powers of \( x \) and \( y \), it means we need to find a constant \( k \) such that \( z = kx^3y^3 \).

• Identify given values to find \( k \).
• Use the formula with known values: \( 2160 = k \cdot 3^3 \cdot 4^3 \).
• Solve for \( k \) by simplifying: \( k = \frac{2160}{27 \cdot 64} \).

This computation shows the central role that the constant of variation plays in forming meaningful equations from stated conditions.

Proportional Relationships

Proportional relationships are fundamental in describing how quantities scale together. In this context, a quantity like \( z \), which varies jointly with \( x^3 \) and \( y^3 \), suggests that any change in these variables will result in a scaled change in \( z \) by a specific multiplicative factor determined by \( k \).
For the joint variation problem, once we've established the equation \( z = kx^3y^3 \), we see how \( z \) changes proportionally with \( x \) and \( y \). This allows us to understand or predict how variations in \( x \) and \( y \) affect \( z \).

• It requires a constant rate of change, \( k \).
• Shows linearity when graphed in a transformed space (e.g., log-log plots).
• Enables solving for unknowns when additional data points are provided.

Understanding these relationships enhances our ability to model real-world scenarios where quantities depend on multiple factors.

Exponents

Exponents are powerful mathematical tools used to express repeated multiplication. In the context of the joint variation, exponents like \( x^3 \) or \( y^3 \) imply that the variable is being multiplied by itself three times.
When solving problems of joint variation, exponents modify the scale at which variables affect the output. For example, in the equation \( z = kx^3y^3 \), both \( x \) and \( y \) affect \( z \) exponentially.\

• Calculate exponents: \( x^3 \) means \( x \times x \times x \).
• Exponents show how small changes in input lead to large changes in output.
• Exponential growth or decay represented compactly.

This concept is crucial for understanding how inputs might influence outputs more dramatically than a simple linear multiplied approach, particularly important in scientific calculations and real-world applications.