jfan17

I've tried posting answers to this before but the explanations were too long so I'll post it in bits:

1.  This is a textbook case of a method of counting known colloquially as "stars and bars" or "balls and urns"(I'd recommend you to search it up online). We can think of the 50 votes as 50 balls laid out in a row, divided by 3 "dividers" which separate the balls into 4 separate regions for each candidate, with each region being how many votes a candidate gets. Using stars and bars, we get the equation:

(50+4-1) c (50),  or (53c50). This gives us 53*52*51/3!

This simplifies to:

53*26*17 = 23426 totals

Oh i think that would explain it

(x+y)^2 = 16

x^2+2xy+y^2 = 16

x^2 + y^2 = 22

2xy = -6

xy = -3

y = -3/x

(x - 3/x)^2 = 16

x^2 - 6 + 9/(x)^2 = 16

x^2 + 9 / (x)^2 = 22

Completing the square, we can get:

x^2 + 6 + 9/(x)^2 = 28

This is useful, because it asks us for x-y. Remembering our earlier substitution of y = -3/x:

x - y = x - (-3/x) = x + 3/x

(x+3/x)^2 = 28

(x+3/x) = sqrt(28)

Let's first start off with a knowledge of what "average speed" actually is. Average speed is defined as:

total distance / total time

The distance to the train station is equal to 45 * 4km, which is how far he traveled on the return trip(45 km per hour for 4 hours). That means that the distance for one trip is equal to 180 km. Since we only need total distance / total time, the total distance is equal to the distance of one trip multiplied by 2, which is 180*2 = 360. Then, total time is just 6 hours + 4 hours, which is 10 hours.

360 / (6+4) = 360 / 10 = 36 km/h

I already posted an answer on this and there are previous questions asking this. Not sure if they've gone through moderation though.

Hey guest! Honestly, in my opinion, as long as you eat enough, then sleep typically > eating. However, in a life or death situation, eating would probably be more important than sleeping(as long as you don't die from sleep deprivation ofc)

Sorry the roots should be -3 and -2 I believe Pavlov

Let's call the lengths of the rectangular prism x, y, and z. The problem tells us that the sum of the edges is 40, so we can write:

x + y + z = 40

The problem also says that the surface area is 48. That means:

2(xy +yz + xz) = 48

Realize that if we square (x + y + z), that is equal to :

(x^2 + y^2 + z^2 + 2xy + 2xz + 2yz), or :

(x^2 + y^2 + z^2) + 2(xy + xz + yz)

(try expanding it out yourself if you want!)

(x^2 + y^2 + z^2 ) + 48 (since the surface area is 48. See how the substitution works?). 

This is then equal to 1600(40 ^2).

Subtracting 48 on both sides, we get:

(x^2 + y^2 + z^2) = 1552

What's important to realize here is that the question asks us for the space diagonal(the diagonal connecting opposite corners of the prism). The general formula for the length of a space diagonal in a rectangular prism is: sqrt(x^2 + y^2 + z^2). This can be derived from using pythagorean theorem if you really want to see for yourself. 

That means our desired diagonal length is sqrt(1552), which is simplifies to : 4sqrt(97) 

Let's start off and name the angle between the hour and minute hand x, representing the number of degrees between the hands. Next, let's look at the minute hand first. After exactly an hour, the minute hand does not change its position(since after one hour, it comes back to where it was previously). As for the hour hand however, its position differs by exactly "1 hour tick", which is equivalent to 30 degrees (360 / 12 hours). What that implies is that x would have to equal x + 30, which is clearly impossible. However, we can think about the problem in a different way: by looking at the acute angle only between the hands, that would indeed be possible. Assuming that x + 30 is greater than 180(since if it wasn't, that wouldn't be possible), we can write the following equation:

x = 360 - (x+30) 

, with the right hand side coming from the fact that the obtuse angle of x + 30 must have an acute counterpart, of which both add up to 360. 

Then, we can simplify to:

x = 330 - x

or:

2x = 330

x = 165.

This gives us our only answer of 165 degrees