The factors of a number are all positive integers that divide evenly into the number. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40.

A ratio of 4:1 means that there are 4 home fans for every one visiting fan. So, of every 5 fans, 4 are home fans and \( \frac{4}{5} \) of every fan in the stadium is a home fan:

41,000 fans x \( \frac{4}{5} \) = \( \frac{164000}{5} \) = 32,800 fans.

The area of a rectangle is width (w) x height (h). In this problem we know that the rectangle is twice as long as it is wide so h = 2w. The perimeter of a rectangle is 2w + 2h and we know that the perimeter of this rectangle is 24 meters so the equation becomes: 2w + 2h = 24.

Putting these two equations together and solving for width (w):

2w + 2h = 24
w + h = \( \frac{24}{2} \)
w + h = 12
w = 12 - h

From the question we know that h = 2w so substituting 2w for h gives us:

w = 12 - 2w
3w = 12
w = \( \frac{12}{3} \)
w = 4

Since h = 2w that makes h = (2 x 4) = 8 and the area = h x w = 4 x 8 = 32 m²

To add these radicals together their radicands must be the same:

6\( \sqrt{50} \) + 6\( \sqrt{2} \)
6\( \sqrt{25 \times 2} \) + 6\( \sqrt{2} \)
6\( \sqrt{5^2 \times 2} \) + 6\( \sqrt{2} \)
(6)(5)\( \sqrt{2} \) + 6\( \sqrt{2} \)
30\( \sqrt{2} \) + 6\( \sqrt{2} \)

Now that the radicands are identical, you can add them together:

To simplify this fraction, first find the greatest common factor between them. The factors of 40 are [1, 2, 4, 5, 8, 10, 20, 40] and the factors of 60 are [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60]. They share 6 factors [1, 2, 4, 5, 10, 20] making 20 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{40}{60} \) = \( \frac{\frac{40}{20}}{\frac{60}{20}} \) = \( \frac{2}{3} \)