If 76% of the town's 36,000 voters cast ballots the number of votes cast is:

(\( \frac{76}{100} \)) x 36,000 = \( \frac{2,736,000}{100} \) = 27,360

The mayor got 57% of the votes cast which is:

(\( \frac{57}{100} \)) x 27,360 = \( \frac{1,559,520}{100} \) = 15,595 votes.

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 36 meters so the equation becomes: 2w + 2h = 36.

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

2w + 2h = 36
w + h = \( \frac{36}{2} \)
w + h = 18
w = 18 - h

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

w = 18 - 2w
3w = 18
w = \( \frac{18}{3} \)
w = 6

Since h = 2w that makes h = (2 x 6) = 12 and the area = h x w = 6 x 12 = 72 m²

To simplify a radical, factor out the perfect squares:

\( \sqrt{32} \)
\( \sqrt{16 \times 2} \)
\( \sqrt{4^2 \times 2} \)
4\( \sqrt{2} \)

The number of students taking German or Spanish is 9 + 11 = 20. Of that group of 20, 5 are taking both languages so they've been counted twice (once in the German group and once in the Spanish group). Subtracting them out leaves 20 - 5 = 15 who are taking at least one language. 24 - 15 = 9 students who are not taking either language.

First, solve for \( \left|a - 8\right| \):

\( \left|a - 8\right| \) - 5 = 8
\( \left|a - 8\right| \) = 8 + 5
\( \left|a - 8\right| \) = 13

The value inside the absolute value brackets can be either positive or negative so (a - 8) must equal + 13 or -13 for \( \left|a - 8\right| \) to equal 13:

So, a = -5 or a = 21.