$a$ varies directly as the square of $b$ and inversely as $c$. Given that $a=2$ when $b=4$ and $c=24$, find the constant of proportionality $k$.

1. Problem Description

$a$ varies directly as the square of $b$ and inversely as $c$. Given that $a=2$ when $b=4$ and $c=24$, find the constant of proportionality $k$.

2. Solution Steps

The problem states that $a$ varies directly as the square of $b$ and inversely as $c$. This can be expressed as:

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

We are given that $a=2$, $b=4$, and $c=24$. We can substitute these values into the equation to solve for $k$:

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

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

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

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

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

$3 = k$

3. Final Answer

The value of $k$ is 3.