Final answer:
The constant of variation (k) is found to be 8 using the given values, and with the constant, we determine that the value of z is 4 when x = 6 and f = 12.
Step-by-step explanation:
When we say that z varies directly with the square of x and inversely with f, we are describing a form of joint variation which can be represented mathematically as z = kx2/f, where k is the constant of variation. To find the value of k, we can use the given values z = 4, x = 2 (since 22 = 4), and f = 8.
To find k, we rearrange the equation to k = zf/x2 and substitute the known values to get k = (4)(8)/22, so k = 32/4 = 8. With k determined, we can now find z when x = 6 and f = 12 using z = kx2/f which gives z = (8)(62)/12 = 48/12 = 4. Therefore, the constant of variation is 8 and the value of z is 4 for the given values of x and f.