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)
Algebraic equation:
z = kyx^3
Substituting:
96 = k(6)(2)^3
96 = 48k
2 = k
Therefore: z = 2yx^3
Certainly! To find the variation equation for this problem, we need to determine how the variables z, y, and x are related.
The variation statement tells us that z varies jointly as y and the cube of x. This means that z is directly proportional to both y and the cube of x.
We can write the variation equation in the form:
z = k * y * (x^3)
Where k is the constant of variation.
To find the value of k, we can substitute the given values of z, x, and y into the equation and solve for k.
Given that z = 96 when x = 2 and y = 6, we can write:
96 = k * 6 * (2^3)
96 = 48k
Now, we can solve for k:
k = 96 / 48
k = 2
So, the value of k is 2.
Substituting the value of k back into the variation equation, we get:
z = 2 * y * (x^3)
Simplifying further, we can rewrite this as:
z = 2xy^3
Therefore, the correct answer is option (b): y = 2xy^3.