BuilderBoi

Extend point P so it is perpendicular to AB. Label the intersection point F. 

\([AMP] = {2 \over3} \times PF \div 2\)

\(\triangle APF \) is similar to \(\triangle ABN\). Because of this, \(PF = {1 \over3}\)

Thus, \([AMP] = {2 \over 3} \times {1 \over 3} \div 2 = \color{brown}\boxed{1 \over 9}\)

Gahh... Speed is tricky

The big square has side length of 5. 

The small square has a side length of \(\sqrt {12} = 2\sqrt3\)

The rectangles are congruent, so their width can be expressed as \(2x+2\sqrt3=5\)

Solving for x, we find \(x = {{5 - 2\sqrt3} \over 2}\)

The length can be expressed as \({2\sqrt3}+{5 -2\sqrt3 \over 2} = {5+2\sqrt3 \over2}\)

Using the Pythagorean Theorem, the length of the diagonal is \(\color{brown}\boxed{5\sqrt2 \over2}\)

Happens to the best of us lol...

Because it's the same distance, it is \((60 + 80) \div 2 = \color{brown}\boxed{70} \) mph. 

Factor the left-hand side of the second equation: \(2a(b+5)=3b+257\)

Do the same thing to the other side: \(2a(b+5)=3(b+5)+242\)

Isolate the constant and simplify: \((2a-3)(b+5)=242\)

The only "reachable" factors of 242 are 22 and 11. 

We can achieve these numbers when \(a = 7\) and \(b = 17\).

Thus, \(ab = 17 \times 7 = \color{brown}\boxed{119}\)

\(x =\)  The number of times GetThere Airlines increases the ticket cost by \($12\)

Write as an equation: \((200+12x)(40,000-500x)\)

Simplify: \(−6000x^2+380000x+8000000\)

The vertex appears at \(-380000\div -12000 = 31 {2 \over 3} \approx 32\)

Thus, they make the most money when the tickets are priced at: \(200 +32 \times 1 2 = \color{brown}\boxed{584}\)

The answer isn't mine tho... 

It's Melody's and I trust them more than I trust myself...

There are 9 ways to choose a single person.

You then have \(8 \choose 4\) ways to choose a group of 4 people. But, you have to divide by 2, to correct for overcounting.

Once that is done, you have \({4 \choose 4} = 1\) way to choose the remaining. 

Thus, there are \(9 \times {8 \choose 2} \div 2 = \color{brown}\boxed{315}\) ways.

All credit goes to Melody: https://web2.0calc.com/questions/a-group-of-nine-people-is-be-split-into-three-teams_1

Ayoo... That's my 400th answer