Final answer:
The question explores a direct variation relationship where y varies with some root of x. A specific case is given, and from that, the constant of proportionality can be determined. With this constant, further values of y corresponding to different x values can be calculated.
Step-by-step explanation:
The question asks to explore the relationship where y varies directly with some root of x, given that when x=4, y=16. To describe this relationship, you can use a direct variation equation of the form y = k * root(x), where 'k' is the constant of proportionality. Given a specific pair of values x=4 and y=16, we can solve for 'k' and establish the specific relationship between y and x.
For example, if y varies directly with the square root of x, the equation becomes y = k * sqrt(x). Substituting the given values 4 and 16, we find that k = 16 / sqrt(4) = 16 / 2 = 8. The relationship is then y = 8 * sqrt(x).
To apply this further, you may be asked to find y for other values of x or determine x when a certain y value is given. It's important to calculate correctly using the constant of proportionality and the correct root of x, whether that's a square root, cube root, etc., depending on the problem's specifics.