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 18 meters so the equation becomes: 2w + 2h = 18.

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

2w + 2h = 18
w + h = \( \frac{18}{2} \)
w + h = 9
w = 9 - h

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

w = 9 - 2w
3w = 9
w = \( \frac{9}{3} \)
w = 3

Since h = 2w that makes h = (2 x 3) = 6 and the area = h x w = 3 x 6 = 18 m²

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [20, 40, 60, 80] making 20 the smallest multiple 4 and 10 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{2 x 5}{4 x 5} \) + \( \frac{3 x 2}{10 x 2} \)

\( \frac{10}{20} \) + \( \frac{6}{20} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{10 + 6}{20} \) = \( \frac{16}{20} \) = \(\frac{4}{5}\)

There are 2 5-passenger taxis available so that's 2 x 5 = 10 total seats. There are 12 people needing transportation leaving 12 - 10 = 2 who will have to find other transportation.

A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{6!}{2!} \)
\( \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} \)
\( \frac{6 \times 5 \times 4 \times 3}{1} \)
\( 6 \times 5 \times 4 \times 3 \)
360

To convert a negative exponent to a positive exponent, calculate the positive exponent then take the reciprocal.