The problem states that $a$ varies directly as the square of $b$ and inversely as $c$. We are given that $a=2$ when $b=4$ and $c=24$. We need to find the constant of proportionality, which we'll call $k$.

1. Problem Description

The problem states that $a$ varies directly as the square of $b$ and inversely as $c$. We are given that $a=2$ when $b=4$ and $c=24$. We need to find the constant of proportionality, which we'll call $k$.

2. Solution Steps

The relationship between $a$, $b$, and $c$ can be expressed as:

$a = k \cdot \frac{b^2}{c}$

We are given $a=2$, $b=4$, and $c=24$. Plugging these values into the equation, we get:

$2 = k \cdot \frac{4^2}{24}$

$2 = k \cdot \frac{16}{24}$

$2 = k \cdot \frac{2}{3}$

To solve for $k$, we multiply both sides of the equation by $\frac{3}{2}$:

$k = 2 \cdot \frac{3}{2}$

$k = 3$

3. Final Answer

The value of $k$ is 3.