Problem 85 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. If \(a\) varies directly with \(m\) and \(n^{2}\) and inversely with \(y^{3}\) and if \(a=9\) when \(m=4, n=9,\) and \(y=3,\) find \(a\) if \(m=6, n=2,\) and \(y=5\)

Step-by-Step Solution

Answer

The value of \( a \) is approximately 0.144.

Step 1: Understand the Variation Relationship

If a quantity varies directly as two or more quantities and inversely as another, we express this as a formula. Here, \( a \) varies directly with \( m \) and \( n^2 \) and inversely with \( y^3 \). Thus, we can write: \[ a = k \frac{m n^2}{y^3} \]where \( k \) is the constant of proportionality.

Step 2: Find the Constant of Proportionality

We're given that \( a = 9 \) when \( m = 4, n = 9, \) and \( y = 3 \). Plug these values into the equation to solve for \( k \):\[9 = k \frac{4 \times 9^2}{3^3}\]Calculate the values inside the equation:\[9 = k \frac{4 \times 81}{27}\]\[9 = k \frac{324}{27}\]\[9 = k \times 12\]Divide both sides by 12 to solve for \( k \):\[k = \frac{9}{12} = \frac{3}{4}\]

Step 3: Calculate the New Value of \( a \)

Now that we have \( k = \frac{3}{4} \), use it to find \( a \) with the new values \( m=6, n=2, y=5 \): \[a = \frac{3}{4} \frac{6 \times 2^2}{5^3}\]Calculate the power and product inside:\[a = \frac{3}{4} \frac{6 \times 4}{125}\]\[a = \frac{3}{4} \frac{24}{125}\]\[a = \frac{3 \times 24}{4 \times 125}\]Simplify the fraction:\[a = \frac{72}{500}\]Reduce \( \frac{72}{500} \) to simplest form:\[a = \frac{36}{250} = \frac{18}{125} = \frac{9}{62.5}\]The approximate numerical value for \( a \) is 0.144.

Key Concepts

Proportionality Constant

When dealing with direct and inverse variation problems, the proportionality constant, often denoted as \( k \), acts as a key multiplier. It helps us connect the variables in a precise way. In mathematical terms, when a variable \( a \) varies directly with some quantities like \( m \) and \( n^2 \), and inversely with another \( y^3 \), it can be formulated as \[ a = k \frac{m n^2}{y^3} \].

Here, \( k \) ensures that the equation correctly represents the relationship between these variables in any configuration. It's important to solve for \( k \) accurately using given values, like when \( a = 9 \), \( m = 4 \), \( n = 9 \), and \( y = 3 \). By substituting these values back into the equation, you can solve for \( k \) mathematically by rearranging the equations and performing arithmetic operations as shown: \( 9 = k \times 12 \) eventually gives us \( k = \frac{3}{4} \). This calculated value then allows us to explore how \( a \) changes when \( m \), \( n \), and \( y \) are altered.

Solving Equations

Solving equations with direct and inverse variation can feel complex, but breaking it into manageable steps makes it easier.

• First, set up the equation that represents the relationship using the given dependencies. In this problem, we know \( a = k \frac{m n^2}{y^3} \).
• Next, use known values to find \( k \), the constant of proportionality, which is key to solving for any other variable in the equation.
• Once \( k \) is determined, apply it to new values. Substitute these new values back into the equation, and simplify step-by-step.
• Accurate calculation is crucial. Pay attention to the arithmetic operations, especially division and multiplication.
• Finally, simplify fractions or expressions to arrive at a clean, understandable final answer.

As seen in the example, identify each component's role and use algebraic manipulation effectively to solve for the unknown. This approach allows you to find solutions in various situations smoothly.

Mathematical Modeling

Mathematical modeling involves creating an equation or a set of equations that represent real-world scenarios effectively. It’s a powerful tool used to predict or analyze situations based on variable relationships.

In the given example, the setup equation \( a = k \frac{m n^2}{y^3} \) models a specific context where a quantity's behavior is influenced by direct and inverse relationships. First, you establish the relationship using an appropriate formula, capturing how changes in certain variables affect others.

The calculated proportionality constant \( k \), derived from initial conditions, ensures that the model is accurate within the context provided by initial data. Then, the equation is used to predict values like determining \( a \) when new values of \( m, n, \) and \( y \) are given.

Through equations and algebra, mathematical models help identify patterns, forecast changes, and solve problems. They are crucial for making informed decisions in science, engineering, economics, and many other fields.