U varies directly as the square root of V when U = 24, V = 9, find the value of V when U = 16.

1. Analyze the Question

The problem involves direct variation. We are told that $U$ varies directly as the square root of $V$. We are given an initial set of values ($U=24, V=9$) to find the constant of variation, and then asked to find $V$ when $U=16$.

2. Explain the Concept

When a variable $U$ varies directly as the square root of $V$, it is expressed mathematically as:

To turn this into an equation, we introduce a constant of proportionality, $k$:

Our goal is to first find $k$ using the initial values, then use that $k$ to solve for the unknown $V$.

3. Show the Solution

Step 1: Find the constant $k$ Using $U = 24$ and $V = 9$:

Since $\sqrt{9} = 3$:

Divide both sides by 3:

Step 2: Find $V$ when $U = 16$ Now that we know $k = 8$, our formula is $U = 8\sqrt{V}$. Substitute $U = 16$:

Divide both sides by 8:

To solve for $V$, square both sides of the equation:

4. Identify the Answer

The value of $V$ is 4. This corresponds to option B.

5. Exam Tip

In variation problems, always find the constant $k$ first. Think of $k$ as the "bridge" that connects the two variables. Once you have $k$, you can solve for any missing value in that relationship. Also, remember that "varies directly" means the variables move in the same direction, while "varies inversely" means they move in opposite directions.