Problem 29 - Intermediate Algebra

Question

If \(A\) varies jointly as \(b\) and \(h\), and \(A=60\) when \(b=12\) and \(h=10\), find \(A\) when \(b=16\) and \(h=14\).

Step-by-Step Solution

Answer

When \( b = 16 \) and \( h = 14 \), \( A = 112 \).

Step 1: Understand Joint Variation

If a variable varies jointly with two other variables, it is equal to a constant multiplied by the product of those variables. The problem states that \( A \) varies jointly as \( b \) and \( h \). This relationship can be written as \( A = kbh \), where \( k \) is the constant of variation.

Step 2: Find the Constant of Variation

Given \( A = 60 \), \( b = 12 \), and \( h = 10 \), substitute these values into the joint variation formula: \( 60 = k \times 12 \times 10 \). Simplifying gives \( 60 = 120k \), so \( k = \frac{60}{120} = \frac{1}{2} \).

Step 3: Substitute Values for New Condition

Now that we know \( k = \frac{1}{2} \), we can find \( A \) when \( b = 16 \) and \( h = 14 \). Substitute these values into the formula: \( A = \frac{1}{2} \times 16 \times 14 \).

Step 4: Calculate the New Area

Calculate the product of \( 16 \) and \( 14 \), which equals \( 224 \). Multiply by \( \frac{1}{2} \) to find \( A = 112 \).

Key Concepts

Constant of Variation

The constant of variation is an important concept when dealing with joint variation problems. It acts as the link between the variables in question. When we say that a variable varies jointly with two others, it means there's a consistent relationship depicted by a constant.

In our exercise, the relationship between the area (\(A\)), base (\(b\)), and height (\(h\)) is expressed through the equation \(A = kbh\), where \(k\) is the constant of variation. From the initial scenario where \(A = 60\), \(b = 12\), and \(h = 10\), we solved for \(k\), finding it to be \(\frac{1}{2}\).

This constant allows us to predict the area (\(A\)) for any combination of \(b\) and \(h\), as long as the relationship remains consistent. Understanding and calculating this constant is crucial to tackling any problems involving variation.

Algebraic Expressions

Algebraic expressions are key to solving joint variation problems. They allow us to represent relationships mathematically and make computations manageable.

In joint variation, our algebraic expression is \(A = kbh\). This denotes that \(A\) is dependent on the product of \(b\) and \(h\) scaled by the constant \(k\).

• To solve for unknowns, substitute known values into the expression.
• Simplify the equation to find the constant or the requested variable.
• Use the expression to predict new outcomes under different scenarios.

Understanding how to rearrange and solve such expressions is foundational in algebra, leading to efficient problem-solving processes.

Problem Solving Steps

Solving problems involving joint variation requires methodical steps to arrive at a solution. Here is a simple breakdown based on the exercise:

1. **Identify the Relationship**: Understand the type of variation involved. In this case, it's joint variation shown by \(A = kbh\).2. **Compute the Constant**: Using given values, substitute them into the joint variation formula to find \(k\). For example, substituting \(A = 60\), \(b = 12\), \(h = 10\) gave \(k = \frac{1}{2}\).3. **Apply the Formula**: With \(k\) determined, substitute new conditions (\(b = 16\), \(h = 14\)) into \(A = kbh\) to find the new \(A\). Each step is integral in logically progressing toward not just a correct answer, but also a comprehensive understanding of how and why the solution works.