If P varies inversely as the square root of q, where p = 3 and q = 16, find the value of q when p = 4.

1. Analyze the Question

We are given a relationship of inverse variation between two variables, $p$ and $q$. Specifically, $p$ varies inversely as the square root of $q$. We are given an initial set of values to find the constant of variation, and then asked to find a new value for $q$ when $p$ changes.

2. Explain the Concept

In mathematics, when $p$ varies inversely as the square root of $q$, the relationship is expressed as:

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

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

3. Show the Solution

Step 1: Find the constant $k$ Given $p = 3$ and $q = 16$:

Since $\sqrt{16} = 4$:

Multiply both sides by 4:

Step 2: Find $q$ when $p = 4$ Now that we know $k = 12$, our formula is $p = \frac{12}{\sqrt{q}}$. Substitute $p = 4$:

Multiply both sides by $\sqrt{q}$ and divide by 4:

To solve for $q$, square both sides:

4. Identify the Answer

Comparing our result to the given options: A. 12 B. 8 C. 9 D. 16

The correct option is C.

5. Exam Tip

When dealing with inverse variation, always remember: "If one goes up, the other goes down." Since $p$ increased from 3 to 4, we should expect $q$ to decrease from its original value of 16. Since 9 is the only option smaller than 16, you could have narrowed your choices down quickly!