Problem 96 - College Algebra with Modeling and Visualization

Question

Solve the variation problem. Suppose \(y\) varies directly as the second power of \(x .\) When \(x=3, y=10.8 .\) Find \(y\) when \(x=1.5\)

Step-by-Step Solution

Answer

When \(x = 1.5\), \(y = 2.7\).

Step 1: Understanding Direct Variation

Since we are dealing with direct variation, we know that if one variable increases, the other variable also increases at a constant proportion. In mathematical terms, if \(y\) varies directly as \(x^2\), we can express this as: \[ y = kx^2 \]where \(k\) is the constant of variation.

Step 2: Finding the Constant of Variation

We are given that when \(x = 3\), \(y = 10.8\). Using the direct variation formula \(y = kx^2\), we can plug in these values to find \(k\): \[ 10.8 = k(3)^2 \]\[ 10.8 = k imes 9 \]\[ k = \frac{10.8}{9} \]\[ k = 1.2 \]

Step 3: Using the Constant to Solve for New \(y\) Value

Now that we have found the constant of variation \(k = 1.2\), we can use it to find \(y\) when \(x = 1.5\). Substitute \(k = 1.2\) and \(x = 1.5\) into the direct variation equation: \[ y = 1.2(1.5)^2 \]Calculate \((1.5)^2 = 2.25\), thus: \[ y = 1.2 imes 2.25 \]

Step 4: Calculating the Final Value

Perform the multiplication to find \(y\): \[ y = 2.7 \] When \(x = 1.5\), \(y\) is \(2.7\).

Key Concepts

Constant of Variation

When dealing with direct variation problems, it's essential to understand the "constant of variation." This constant, often represented by the letter \(k\), plays a crucial role in expressing the relationship between two variables—like \(y\) and \(x^2\) in the problem given. To find the constant of variation \(k\), you need to have initial values given for both variables. In our example, when \(x = 3\) and \(y = 10.8\), we use these to determine \(k\) with the formula \(y = kx^2\). By substituting \(x = 3\) and \(y = 10.8\), it's calculated as \(k = \frac{10.8}{9} = 1.2\).

• This constant indicates how much \(y\) changes when \(x\) is altered.
• It stays the same as long as the situation doesn't change (like different slopes or scales).

Finding \(k\) is always a fundamental step in solving problems involving direct variation.

Quadratic Relationship

In many mathematical scenarios, understanding the type of relationship between variables is vital. Here, we’re looking at a quadratic relationship, as \(y\) varies directly as \(x^2\). This implies that as \(x\) changes, \(y\) changes at a rate proportional to the square of \(x\).Quadratic relationships are nonlinear, which means they don't form a straight line when graphed. Instead, they form a parabola.

• For a deeper understanding, remember that quadratic terms like \(x^2\) imply that small changes in \(x\) can result in much larger changes in \(y\).
• Such relationships are common in physics and engineering, where the effect of a variable accelerates over time.

Recognizing a quadratic relationship helps in visualizing and predicting how changes in one variable will impact the other, vital for problem-solving.

Solving Equations

Solving equations is the process of finding the unknowns in an equation using known values and operations. To solve the exercise, we needed to use the constant of variation within the given equation \(y = kx^2\). It involves substituting the values we know, simplifying, and finding the resulting \(y\) value.For example, after finding \(k = 1.2\), we can substitute \(k\) and \(x = 1.5\) in the equation to find \(y\). The steps include:

• Substitute the known values: \(y = 1.2(1.5)^2\).
• Calculate the square: \((1.5)^2 = 2.25\).
• Solve for \(y\): \(y = 1.2 \times 2.25 = 2.7\).

This method highlights the systematic approach needed to solve mathematical problems, ensuring each step is executed correctly and our solutions are reliable. Mastering this process will enhance confidence in tackling various math problems.