Problem 28 - Algebra and Trigonometry

Question

Express the statement as an equation. Use the given information to find the constant of proportionality. \(M\) is jointly proportional to \(a, b,\) and \(c\) and inversely proportional to \(d\) If \(a\) and \(d\) have the same value and if \(b\) and \(c\) are both \(2,\) then \(M=128 .\)

Step-by-Step Solution

Answer

The constant of proportionality is \(k = 32\).

Step 1: Understand the Proportionality Statement

According to the problem, \(M\) is jointly proportional to \(a, b,\) and \(c,\) and inversely proportional to \(d\). This can be expressed as: \(M = k \cdot \frac{a \cdot b \cdot c}{d}\) where \(k\) is the constant of proportionality.

Step 2: Substitute Given Values

We are given that \(a = d\), \(b = 2\), \(c = 2\), and \(M = 128\). Substitute these values into the equation: \[ 128 = k \cdot \frac{a \cdot 2 \cdot 2}{a} \] This simplifies to \(128 = k \cdot 4\) since \(a\) and \(d\) cancel each other out.

Step 3: Solve for the Constant of Proportionality

To find \(k\), solve the equation from Step 2: \[ 128 = k \cdot 4 \] Divide both sides by 4 to isolate \(k\): \[ k = \frac{128}{4} = 32 \] Thus, the constant of proportionality is \(k = 32\).

Key Concepts

Constant of Proportionality

Understanding the constant of proportionality is key to solving equations involving relationships between variables. This constant, often denoted as \( k \), acts as a scaling factor. It helps balance how one variable changes in response to changes in others. In our situation, the problem describes \( M \) as being jointly proportional to \( a, b, \) and \( c \), and inversely proportional to \( d \).

Here, the constant of proportionality \( k \) connects the product of \( a, b, \) and \( c \) with the value of \( M \). Meanwhile, the inverse relation with \( d \) is shown by dividing the product by \( d \). When finding \( k \), we first restructure the equation \( M = k \cdot \frac{a \cdot b \cdot c}{d} \) to include all given values. Solving for \( k \) involving these specific values confirms the direct effect it has in balancing the equation.

Inversely Proportional

When a quantity is inversely proportional to another, increasing one results in a decrease of the other. In this context, \( M \) is inversely proportional to \( d \). This relationship is mathematically expressed by dividing by \( d \) in the joint proportionality equation.

If two quantities are inversely proportional, their product remains a constant. Therefore, as \( d \) grows, the resultant \( M \) decreases, assuming all other factors remain unchanged. Understanding inverse proportionality is crucial for grasping how the balance in the equation modifies with changes in involved variables.

Mathematical Equations

Mathematical equations are powerful tools for representing relationships between variables. They use mathematical symbols and operations to convey complex information in a structured form. In this problem, the relationship is encapsulated in the equation \( M = k \cdot \frac{a \cdot b \cdot c}{d} \).

The variables \( a, b, c, \) and \( d \) along with the constant \( k \) are integral to defining how \( M \) behaves. By substituting known values into the equation, you can solve for unknowns, such as the constant of proportionality \( k \).

• The equation balances proportionality, mixing both direct and inverse relationships.
• Careful manipulation of these equations ensures accurate solutions.

Understanding and using equations efficiently is a fundamental skill in mathematics, providing a clear method to solve problems by connecting different mathematical concepts.