A quantity x varies directly with the cube of y and inversely with the square root of z. If x = 457.33333333333 when y = 7 and z = 9, find x when y = 6 and z = 4.

x = ky^3/√z

(x√z)/y^3 = k, a constant.
So, you want x such that
(x√4)/6^3 = (1372/3)(√9)/7^3
...

To solve this problem, we need to use the given relationship between x, y, and z. Let's break it down step by step.

We are told that "a quantity x varies directly with the cube of y." In mathematical terms, this can be expressed as:
x ∝ y^3

Next, we are also told that x varies "inversely with the square root of z." In mathematical terms, this can be expressed as:
x ∝ 1/√z or x ∝ z^(-1/2)

Combining these two relationships, we can write the equation for x as:
x = k * (y^3) * (z^(-1/2))

Now, let's find the value of the constant k. We are given that x = 457.33333333333 when y = 7 and z = 9. Substituting these values into the equation, we get:
457.33333333333 = k * (7^3) * (9^(-1/2))

To find k, we can rearrange the equation and solve for it:
k = 457.33333333333 / [(7^3) * (9^(-1/2))]

Now that we have the value of k, we can use it to find x when y = 6 and z = 4.
Substituting these values into the equation, we get:
x = k * (6^3) * (4^(-1/2))

Calculating x using the value of k and the given values of y and z gives us the final answer.