To find the quadratic function related to the square root function \( k(x) = 5\sqrt{x} \), we first note the relationship between the square root function and the quadratic function.
The square root function \( y = \sqrt{x} \) can be rewritten in terms of the quadratic function, which is \( y = x^2 \). To express \( k(x) = 5\sqrt{x} \) in terms of a quadratic function, we can square both sides of the equation.
Starting with: \[ y = 5\sqrt{x} \]
Squaring both sides gives: \[ y^2 = (5\sqrt{x})^2 \] \[ y^2 = 25x \]
Rearranging this, we can express \( x \) in terms of \( y \): \[ x = \frac{y^2}{25} \]
If we consider \( x \) as a function of \( y \), we can identify the related quadratic function: \[ m(y) = \frac{1}{25}y^2 \]
Since the output of the quadratic function typically corresponds to \( m(x) \) in the original variable, we rewrite this as: \[ m(x) = \frac{1}{25}x^2 \]
Thus, the quadratic function related to the square root function \( k(x) = 5\sqrt{x} \) is: \[ \boxed{\frac{1}{25}x^2} \]