Could you please help me with the following question?

Find the variation equation for the variation statement.
31. z varies jointly as y and the cube of x; z=96 when x=2 and y=6 (I don't understand this one. My answer choices are: a.y=2x^3y,b.y=2xy^3, c.y=-2x^3y, or d. y=-2xy^3)

Sure! To find the variation equation, we need to understand what it means for z to vary jointly as y and the cube of x.

When a variable varies jointly as two or more other variables, it means that it is directly proportional to the product of those variables raised to certain powers. In this case, z varies jointly as y and the cube of x, so we can express this relationship as:

z = k * y * x^3

where k is the constant of variation.

To find the value of k, we can use the given information that z=96 when x=2 and y=6. Plugging these values into the equation, we get:

96 = k * 6 * 2^3
96 = k * 6 * 8
96 = 48k

To solve for k, divide both sides by 48:

96/48 = k
2 = k

So the value of k is 2.

Now, we can substitute the value of k back into our equation to get the variation equation:

z = 2 * y * x^3

Simplifying the equation gives us the final variation equation:

z = 2xy^3

Therefore, the correct answer choice is b) y = 2xy^3.