Let's analyze the changes step by step:

1. If w is quadrupled, the new value of w will be 4w. This will multiply q by a factor of 4.

2. If m is doubled, the new value of m will be 2m. This will divide q by a factor of 2.

3. We don't know what happened to z, but we know the overall factor by which q is multiplied is 0.2222222222222222.

Let's put these changes into the formula for q:

Original q: q = 5w / (4m(z^2))

New q: q' = 5(4w) / (4(2m)(z'^2))

Now, let's simplify the new q:

q' = (20w) / (8m(z'^2))
q' = (5w) / (2m(z'^2))

Now, we compare the new q to the original q to find the overall factor:

q' / q = [(5w) / (2m(z'^2))] / [(5w) / (4m(z^2))]
q' / q = [(5w) * (4m(z^2))] / [(2m(z'^2)) * (5w)]
q' / q = (4m(z^2)) / (2m(z'^2))
q' / q = 2 * (z^2) / (z'^2)

We know that q' / q = 0.2222222222222222, so:

2 * (z^2) / (z'^2) = 0.2222222222222222

Now, let's solve for (z'^2):

(z'^2) = 2 * (z^2) / 0.2222222222222222

To simplify the fraction, we can recognize that 0.2222222222222222 is 2/9:

(z'^2) = 2 * (z^2) / (2/9)
(z'^2) = (z^2) * (9/2)
(z'^2) = (9/2) * (z^2)

Now, let's find the change in z by taking the square root of both sides:

z' = sqrt((9/2) * (z^2))
z' = z * sqrt(9/2)
z' = z * sqrt(4.5)
z' = z * sqrt(9/2)
z' = z * 3/sqrt(2)
z' = z * (3/sqrt(2))

So, z has been multiplied by 3/sqrt(2) or approximately 2.1213203435596424 (since sqrt(2) ≈ 1.4142135623730951).
변화를 단계별로 분석해 보겠습니다:

1. w가 4배가 되면 w의 새 값은 4w가 됩니다. 그러면 q에 4의 계수가 곱해집니다.

2. m이 두 배가 되면 m의 새 값은 2m가 됩니다. 그러면 q를 2의 계수로 나눕니다.

3. z에 무슨 일이 일어났는지 모르지만 q에 곱해지는 전체 계수는 0.222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222