Problem 10 - College Algebra

Question

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies jointly as \(m\) and the square of \(n\) and inversely as \(p\) \(y=15\) when \(m=2, n=1,\) and \(p=6 .\) Find \(y\) when \(m=3, n=4,\) and \(p=10\)

Step-by-Step Solution

Answer

The value of \(y\) when \(m=3, n=4,\) and \(p=10\) is \(216\).

Step 1: Identify the Formula

The problem states that \(y\) varies jointly as \(m\) and the square of \(n\) and inversely as \(p\). This can be translated into the mathematical formula \(y = k * m * n^2 / p\) where \(k\) is the constant of variation.

Step 2: Find the Constant of Variation

Substitute the initial values into the equation to find the constant of variation, \(k\). \(15 = k * 2 * 1^2 / 6\). After doing the math, you obtain \(k = 45\).

Step 3: Substitute the New Values

Next, substitute the new values of \(m\), \(n\), and \(p\) into the equation: \(y = 45 * 3 * 4^2 / 10\). Solve the equation to find the new value of \(y\).

Key Concepts

Joint Variation

Understanding joint variation involves recognizing how one variable changes concurrently in direct proportion to two or more other variables. In mathematical terms, if a variable z varies jointly with variables x and y, we can express this relationship as z = kxy, where k is the constant of variation.

Joint variation can be observed in various real-world phenomena. For instance, the volume of an ideal gas varies jointly with the amount of gas and temperature, and inversely with pressure. Learning to recognize joint variation is essential for analyzing complex relationships between multiple factors.

Inverse Variation

Inverse variation signifies a relationship where an increase in one variable leads to a proportional decrease in another and vice versa. The critical aspect of an inverse variation is the product of the two variables involved remains constant. This relationship is typically represented as y = k/x, where y is inversely proportional to x, and k is the constant of variation.

For those grappling with the concept, consider a seesaw: the distance you sit from the center determines your relative up-and-down motion, representing an inverse relationship. As the distance increases, the effort required to lift you decreases.

Constant of Variation

The constant of variation is a fixed number that relates two variables that are directly or inversely proportional to each other. It can be thought of as the 'multiplier' that determines the strength of the relationship. In an equation like y = kx (direct variation) or y = k/x (inverse variation), the k represents this constant.

Mathematically, solving for the constant is crucial. Once you identify the constant using known values, you can predict how variables will respond to changes elsewhere in the system. For learners, pinpointing and understanding the constant of variation sharpens their problem-solving abilities.

Solving Algebraic Equations

The challenge of solving algebraic equations is a vital skill in mathematics. The process requires manipulating the equation by performing operations that simplify or isolate the variable of interest. It's paramount to maintain the balance of the equation—what you do to one side must be done to the other.

Solving equations often involves several steps—distributing, combining like terms, and using inverse operations to get the variable alone on one side. Each step should move you gradually closer to discovering the value of the unknown variable. As students become more adept at solving these equations, they cultivate a logical, methodical approach to problem-solving in algebra and beyond.