The area of a circle is given by the formula A = πr² where r is the radius of the circle. The radius of a circle is its diameter divided by two so A = π(\( \frac{d}{2} \))². If the diameter of the logo increases by 35% the radius (and, consequently, the total area) increases by \( \frac{35\text{%}}{2} \) = 17\(\frac{1}{2}\)%

An integer is any whole number, including zero. An integer can be either positive or negative. Examples include -77, -1, 0, 55, 119.

Use PEMDAS (Parentheses, Exponents, Multipy/Divide, Add/Subtract):

5 + (3 + 3) ÷ 5 x 3 - 3²
P: 5 + (6) ÷ 5 x 3 - 3²
E: 5 + 6 ÷ 5 x 3 - 9
MD: 5 + \( \frac{6}{5} \) x 3 - 9
MD: 5 + \( \frac{18}{5} \) - 9
AS: \( \frac{25}{5} \) + \( \frac{18}{5} \) - 9
AS: \( \frac{43}{5} \) - 9
AS: \( \frac{43 - 45}{5} \)
\( \frac{-2}{5} \)
-\(\frac{2}{5}\)

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 6 meters so the equation becomes: 2w + 2h = 6.

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

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

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

w = 3 - 2w
3w = 3
w = \( \frac{3}{3} \)
w = 1

Since h = 2w that makes h = (2 x 1) = 2 and the area = h x w = 1 x 2 = 2 m²

A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.