To find the related square root function \( p(x) \) for the quadratic equation \( h(x) = 4x^2 \), we need to express this in terms of a square root function.
Given that \( h(x) = 4x^2 \), we can express \( x \) in terms of \( h(x) \):
\[ h(x) = 4x^2 \]
To find the square root, we take:
\[ x^2 = \frac{h(x)}{4} \]
Taking the square root of both sides gives:
\[ x = \sqrt{\frac{h(x)}{4}} = \frac{1}{2} \sqrt{h(x)} \]
Rearranging gives us:
\[ h(x) = 4x^2 \implies x = \frac{1}{2} \sqrt{h(x)} \]
To formulate this in terms of \( p(x) \) where \( p(x) = \sqrt{ax} \), we can derive \( p(x) \) for the function where \( x \) is a function of \( h \). Starting with \( h(x) \):
Setting \( p(x) = \sqrt{ax} \) means we want to find \( a \) such that \( a \) relates our original quadratic function. We need to find \( p(h^{-1}(x)) \):
- Rearranging \( h(x)= 4x^2 \) gives:
\[ x = \frac{1}{2} \sqrt{h(x)} \]
- Therefore, we can express \( h(x) \) in terms of \( x \) using:
\[ p(x) = \sqrt{4x} \]
which can be expressed as
\[ p(x) = \sqrt{4} \sqrt{x} = 2\sqrt{x} \]
Thus, for the form \( p(x) = \sqrt{ax} \), we identify \( a \) as:
\[ \sqrt{a} = 2 \implies a = 4 \]
Thus, the value of \( a \) is
\[ \boxed{4} \]