35% of the water consumption is industrial use and 21% is personal use so (35% - 21%) = 14% more water is used for industrial purposes. 45,000 gallons are consumed daily so industry consumes \( \frac{14}{100} \) x 45,000 gallons = 6,300 gallons.

The equation for this sequence is:

a_n = a_n-1 + 2(n - 1)

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 2(6 - 1)
a₆ = 21 + 2(5)
a₆ = 31

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 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80]. The first few multiples they share are [8, 16, 24, 32, 40] making 8 the smallest multiple 4 and 8 share.

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

\( \frac{9 x 2}{4 x 2} \) + \( \frac{6 x 1}{8 x 1} \)

\( \frac{18}{8} \) + \( \frac{6}{8} \)

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

\( \frac{18 + 6}{8} \) = \( \frac{24}{8} \) = 3

A ratio of 3:1 means that there are 3 home fans for every one visiting fan. So, of every 4 fans, 3 are home fans and \( \frac{3}{4} \) of every fan in the stadium is a home fan:

32,000 fans x \( \frac{3}{4} \) = \( \frac{96000}{4} \) = 24,000 fans.

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 75% the radius (and, consequently, the total area) increases by \( \frac{75\text{%}}{2} \) = 37\(\frac{1}{2}\)%