A varies directly as b$^2$ when A = 4, b = 1. Find A when b = 2

1. Analyze the question

We are given a relationship of direct variation where $A$ depends on the square of $b$ ($b^2$). We are provided with an initial set of values to find the constant of variation, and then asked to calculate a new value for $A$ when $b$ changes.

2. Explain the concept

In direct variation, the relationship is expressed as:

Where:

• $A$ is the variable.
• $b^2$ is the square of the second variable.
• $k$ is the constant of variation (a fixed number we must find first).

3. Show the solution

Step 1: Find the constant ($k$) Using the values $A = 4$ and $b = 1$:

Step 2: Use the constant to find the new $A$ Now that we know $k = 4$, our formula is $A = 4b^2$. We need to find $A$ when $b = 2$:

4. Identify the answer

Comparing our result to the given options: A. 12 B. 14 C. 16 D. 11

The correct option is C.

5. Tip

When dealing with variation questions, always remember the "Find $k$ first" rule. Once you have the constant of variation, the rest of the problem is just simple substitution. Also, don't forget to square the value of $b$ before multiplying by $k$—many students mistakenly multiply $k$ by $b$ first and then square the result, which leads to the wrong answer!