How can we find the value of z if x varies jointly as y and z, given that x is 8 when y is 4 and z is 9 and we need to find z when x is 16 and y is 6?

To solve this problem, we start with the concept of joint variation. The relationship can be expressed with the formula:

x = k imes y imes z

where k is the constant of variation.

We know from the question that when x = 8, y = 4, and z = 9. We can substitute these values into the joint variation equation to find k.

So, we have:

8 = k imes 4 imes 9

This simplifies to:

8 = 36k

Now, to isolate k, we divide both sides by 36:

k = 8 / 36

Reducing the fraction gives us:

k = 2 / 9

Now that we have the value of k, we can find z when x = 16 and y = 6.

We substitute the known values into the joint variation equation:

16 = (2/9) imes 6 imes z

First, compute (2/9) imes 6:

(2/9) imes 6 = 12 / 9 = 4 / 3

Now, we rewrite the equation:

16 = (4/3) imes z

To find z, multiply both sides of the equation by the reciprocal of (4/3), which is (3/4):

z = 16 imes (3/4)

Calculating this gives:

z = 16 imes 0.75 = 12

The value of z when x = 16 and y = 6 is 12.