Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the square root of \(x .\) When \(x=64,\) then \(y=12 .\) Find \(y\) when \(x=36\).

Step-by-Step Solution

Answer

When \(x=36\), \(y=16\).

Step 1: Understand the Inverse Variation Relationship

When a quantity varies inversely with another, their product remains constant. Since \(y\) varies inversely with the square root of \(x\), it can be expressed mathematically as: \( y = \frac{k}{\sqrt{x}} \), where \(k\) is a constant.

Step 2: Find the Constant of Variation

Use the given values \(x=64\) and \(y=12\) to find \(k\). Thus, \(12 = \frac{k}{\sqrt{64}}\). Substituting \(\sqrt{64} = 8\), we get \(12 = \frac{k}{8}\). Multiplying both sides by 8, we find \(k = 96\).

Step 3: Use the Constant to Find the Unknown Value of y

Now use the constant \(k = 96\) to find \(y\) for \(x = 36\). Substitute into the formula: \( y = \frac{96}{\sqrt{36}} \). Because \(\sqrt{36} = 6\), we have \( y = \frac{96}{6} \). Hence, \( y = 16 \).

Key Concepts

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 64 is 8, because 8 times 8 equals 64. Similarly, the square root of 36 is 6, since 6 times 6 is 36. Square roots are vital in various mathematical concepts, including inverse variation problems. When we talk about a number's square root, we usually refer to its principal (or positive) square root. This is because each positive real number actually has two square roots: a positive and a negative. However, for most practical applications, especially in inverse variation problems, we focus on the positive square root. Understanding square roots is crucial in solving algebraic expressions where they appear. For example, when determining the inverse variation of a variable with the square root of another, you need to be comfortable calculating square roots to find unknown values.

Constant of Variation

The constant of variation is a crucial part of understanding relationships in inverse variation scenarios. In the formula for inverse variation, \( y = \frac{k}{\sqrt{x}} \), the letter \( k \) represents this constant. It stays the same as long as the inverse relationship between \( y \) and the square root of \( x \) remains true.To find the constant of variation, you use known values of \( x \) and \( y \). For example, when given \( x = 64 \) and \( y = 12 \), you can substitute these into the inverse variation formula to solve for \( k \). By calculating, you'll see that \( k = 12 \times 8 = 96 \), where 8 is the square root of 64.Once determined, this constant helps you calculate unknown values of \( y \) for other values of \( x \). The constant of variation essentially "links" the values involved in the relationship, ensuring that the inverse variation holds true across different conditions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They are the backbone of most mathematical problem-solving processes. In an inverse variation situation, our algebraic expression takes the form of \( y = \frac{k}{\sqrt{x}} \).Such expressions facilitate the calculation of unknown values by using known variables and constants. In algebra, variables like \( x \) and \( y \) are placeholders that can represent various numbers, allowing flexibility and adaptability in problem-solving.Algebraic expressions become even more important when you need to substitute values and manipulate the expression to find unknowns. In our problem, for example, once the constant \( k \) is found, substituting \( x = 36 \) helped directly solve for \( y \) efficiently, resulting in \( y = 16 \).By forming and manipulating these expressions, students can solve complex problems by breaking down equations into more straightforward, manageable parts. This process helps develop a deeper understanding of mathematical relationships.