Question

If \(a>0, b>0,\) and \(2 c=\sqrt{\frac{a}{b}}\), what is the value of \(b ?\) (1) \(a=8\) and \(c=2\) (2) \(\frac{c^{2}}{a}=\frac{1}{2}\)

Short answer

b = \(\frac{1}{2}\)

Step-by-step solution

Express given relationship

Given: \(2c = \sqrt{\frac{a}{b}}\). Isolate b: \(2c = \sqrt{\frac{a}{b}}\) Square both sides: \(4c^{2} = \frac{a}{b}\) Express b in terms of a and c: \(b = \frac{a}{4c^{2}}\)

Use data from (1)

Given (1): \(a = 8\) and \(c = 2\). Substitute into simplified equation: \(b = \frac{8}{4(2)^{2}}\) Calculate the value: \(b = \frac{8}{4 \cdot 4} = \frac{8}{16} = \frac{1}{2}\)

Verify using condition (2)

Given (2): \(\frac{c^{2}}{a} = \frac{1}{2}\). Use values: \(\frac{2^{2}}{8} = \frac{4}{8} = \frac{1}{2}\). Condition verified as true.

Key concepts

algebraic equations

Algebraic equations are fundamental in solving word problems. They allow us to express complex relationships between different quantities using mathematical symbols. In the exercise above, the equation given is: \[ 2c = \sqrt{\frac{a}{b}} \]. We use this to find the value of one variable in terms of others by manipulating the equation. By squaring both sides, a common algebraic technique, we can simplify the equation to: \[ 4c^2 = \frac{a}{b} \]. This equation is derived from the original relationship and allows us to express b in terms of a and c.

variables and expressions

Understanding variables and expressions is crucial when working with algebraic equations. Variables like a, b, and c represent unknown values that we aim to find. In this problem, we need to determine the value of b based on given values for a and c. Once we simplified the equation \[ b = \frac{a}{4c^2} \]we replaced a and c with their given values in step 1: \[ a = 8 \] and \[ c = 2 \]. To find b, substitute these values into the simplified equation: \[ b = \frac{8}{4(2)^2} \]. It's important to follow each substitution carefully to avoid errors.

problem-solving steps

Solving algebra word problems involves systematic problem-solving steps. Let's break it down:

• 1. **Express the given relationship:** Start by writing out the relationship between the given quantities using an equation. In our example, that is: \[ 2c = \sqrt{\frac{a}{b}} \].
• 2. **Isolate the target variable:** Manipulate the equation to solve for the desired variable. Here, we isolated b: \[ 4c^2 = \frac{a}{b} \]and then expressed b as: \[ b = \frac{a}{4c^2} \].
• 3. **Substitute given values:** Use the provided information to find the value of the variable. With given values \(a = 8\) and \(c = 2\), we substituted them into our equation: \[ b = \frac{8}{4(2)^2} = \frac{8}{16} = \frac{1}{2} \].
• 4. **Verify with additional data:** Use any other provided data to check your results. In our solution, the verification step was crucial. We were given: \[ \frac{c^2}{a} = \frac{1}{2} \].By substituting \(a = 8\) and \(c = 2\): \[ \frac{2^2}{8} = \frac{4}{8} = \frac{1}{2} \].The condition is satisfied, confirming our solution is correct.

Breaking down problems into these steps ensures clarity and accuracy, making it easier to solve complex algebraic problems.