Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, use the given information to find the unknown value. \(y\) varies directly as the square root of \(x\). When \(x=16,\) then \(y=4 .\) Find \(y\) when \(x=36\)

Step-by-Step Solution

Answer

When \(x=36\), \(y=6\).

Step 1: Understanding Direct Variation with Square Root

The statement "\(y\) varies directly as the square root of \(x\)" indicates that there is a direct relationship between \(y\) and \(\sqrt{x}\). This can be expressed mathematically as \(y = k\sqrt{x}\), where \(k\) is a constant.

Step 2: Find the Constant of Variation

To find \(k\), use the given values: when \(x = 16\), \(y = 4\). Substitute these values into the equation \(y = k\sqrt{x}\): \(4 = k\sqrt{16}\). The square root of 16 is 4, so substitute to find \(k\): \(4 = k \times 4\). Solve for \(k\) to get \(k = 1\).

Step 3: Substitute Back into the Formula

Now that we know \(k = 1\), we can rewrite the direct variation equation as \(y = \sqrt{x}\).

Step 4: Calculate \(y\) When \(x = 36\)

Find \(y\) when \(x = 36\) using the equation \(y = \sqrt{x}\). Substitute \(x = 36\) into the equation: \(y = \sqrt{36}\). The square root of 36 is 6, which means \(y = 6\).

Key Concepts

Understanding Square Roots

To understand how direct variation involves square roots, let's first explore what a square root is. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because \(4 \times 4 = 16\). Similarly, the square root of 36 is 6 because \(6 \times 6 = 36\). When looking at relationships, especially direct ones, we often see square roots involved because they simplify the relationship between numbers and can represent non-linear relationships.
This is particularly useful in equations where you need to find a value when a variable has changed. Understanding square roots can help simplify complex equations, making them easier to solve.

The Importance of Constant of Variation

The constant of variation, represented by \(k\) in many mathematical equations, is a crucial factor in understanding direct variation. In our example, you find this constant by looking at the relationship between \(y\) and \(\sqrt{x}\).
Direct variation means that as one variable changes, the other does so in a predictable manner, defined by this constant. If \(y = k\sqrt{x}\), the equation clearly shows that \(y\) scales by \(k\) times the square root of \(x\).
In our specific example, with the values \(y = 4\) when \(x = 16\), we calculated \(k\) to be 1. This means in this relationship, \(y\) is purely the square root of \(x\) with no additional scaling factor. Finding the constant is important as it defines the specific relationship and allows us to predict unknown values.

Exploring Mathematical Relationships

Mathematical relationships, like the direct variation portrayed here, are about understanding how different quantities relate within equations. This relationship involves one variable changing predictably with another. In this case, \(y\) varies directly with \(\sqrt{x}\).
These relationships are often expressed in formulaic terms, using expressions and equations to illustrate how the numbers are linked. They help us predict outcomes (like finding \(y\) when \(x\) changes) and understand the proportional nature of the variables. A direct relationship means that the ratio between the two values remains constant, governed by the constant of variation \(k\).
Understanding these relationships not only helps with this specific problem but also provides a foundation for tackling more complex problems in mathematics, where variables can have more intricate interdependencies.