x varies directly as the square root of y and the square of z. Given that x = 9 when y = 144 and z = 4 , find the value of y when x = 27 and z = 6.

To solve this problem, we first need to understand the relationship between x, y, and z. Since x varies directly as the square root of y and the square of z, we can write this relationship as:

x = k * sqrt(y) * z^2

where k is the constant of variation.

Step 1: Find the constant of variation (k) We know that x = 9 when y = 144 and z = 4. Substituting these values into the equation gives:

9 = k * sqrt(144) * 4^2 9 = k * 12 * 16 9 = 192k k = 9 / 192 k = 0.046875

Step 2: Find the value of y when x = 27 and z = 6 Substitute x = 27, z = 6, and k = 0.046875 into the equation:

27 = 0.046875 * sqrt(y) * 6^2 27 = 0.046875 * sqrt(y) * 36 27 = 1.6875 * sqrt(y) sqrt(y) = 27 / 1.6875 sqrt(y) = 16 y = 16^2 y = 256

So, the value of y when x = 27 and z = 6 is 256.