CentsLord

Let's say Anna finished in 1000 seconds (1 meter per second). Therefore, she finished in 1000 seconds.

Bob's speed is 80% of Anna's, as he had run 80% of the distance Anna did in the same amount of time. Therefore, his speed is 0.8 meters per second.

Using the same logic, Carol's speed is 0.6 meters per second.

Bob must've finished in 1000 / 0.8 = 1250 seconds. 

Going back to Carol. From the beginning of the race to when Bob finished (after 1250 seconds), Carol must have gone 0.6 m * 1250 sec = 750 meters, behind Bob by 1000 - 750 = 250 m.

So, Carol was behind Bob by 250 m when he finished :D

The hero we need AND deserve :D

Trig

So, we have a right triangle. 35 feet is the measure of the vertical leg, and 75 ft is the measure of the other leg.

The tangent of the angle of elevation would be \(\frac{35}{75}\) . So, tan(x) = 35/75 = 7/15, and 7/15 = arctan(x) + pi with the reference angle, that simplifies to something like 25.0165 degrees + pi with the R.A. I am not the best at this stuff :(

So, I'm just going to randomly calculate the hypotenuse ¯\_(ツ)_/¯

Using the Pythagorean Theorem, we know that \(35^2 \cdot 75^2 = \text{hypotenuse}^2 = 6850\)

Therefore, the hypotenuse of the triangle is the square root of 6850. I just found this to complete the triangle, it's unnecessary :D

How did you know the mixed number is 24/9? I don't think some images are showing up on my computer, I couldn't see it D:

6 * 1/x = x

6/x = x

6 = x²

6 = sqrt(6)

Of course, this is assuming 6 times the reciprocal of a number is the number itself. What's in the gap? :D

True, but from exactly a year ago!

Coincidence? o _ O

BTW, the web2.0calc LaTeX/TeXer does not need dollar signs :D

You already asked this question, is more explanation needed? :D

TY LuckyDucky!

Some people may be able to figure it out in a few seconds, but we can use some variables to solve it. Let's call them  "a" and "b".

We know that ab = -35, and a + b = -6

We can express one variable in terms of another, to reduce into a one-variable equation which is easy to solve!

We could write a = -6 -b, or b = -6 - a. Let's go with the first one.

(-6 - b)b = -35

-6b - b² = -35

Wow, it's a quadratic (has an exponent of 2 on the variable) :D 

We can rearrange this into the standard form for quadratic equations (ax² + bx + c = 0).

-b² - 6b + 35 = 0. 

Using any method of solving quadratic equations, we get some really weird numbers for a and b that still fit:

\(-3 - 2\sqrt{11} \) and also \(2\sqrt{11} - 3\)

So, well... there's your answer! :D