Final answer:
To find the joint variation of z with y and the cube of x, establish the equation z = kxy³ and use the given values to solve for the constant k. When x=2 and y=-6, z equals 336, from which we find k = -7.
Step-by-step explanation:
The question asks to establish a relationship where z varies jointly as y and the cube of x. This means that z = kxy³, where k is the constant of proportionality. We are given that z = 336 when x = 2 and y = -6. To find the constant k, we can substitute these values into the equation: 336 = k(-6)(2)³. Simplifying, we get k = 336 / (-6)(8), which results in k = -7. Now that we have the constant, the equation modeling the relationship is z = -7xy³.