proyaop

The total number of ways to get from (0,0) to (4,6) as learned in basic counting is 4 + 6 choose 4. 10 choose 4 = 210.

The number of ways to get from (0,0) to (2,4) is 6 choose 2, and the ways to get from (2,4) to (4,6) is 4 choose 2. Total ways is 15 * 6 = 90

Why calculate that? So now we can know the total number of ways to get their, and the total number of ways to get from the origin to (4,6) that pass through the frog and subtract the two totals. 

210 - 90 = 120 total ways Marvin can reach his home.

2n/n = 2

2 - 2n/(n + 1) = f(n) * 2

1 - n/(n + 1) = f(n)

n > or = to 0.

1 - 0 = f(0)

1/2 = f(1)

1/3 = f(2)

1 - 3/(3 + 1) = 1/4 = f(3)

So you can see the pattern, and thus f(n) = 1/(n + 1)

First row is 1

then 1,1

then 1, 2, 1

then 1, 3, 3, 1 = average of 2. n = 3

The equation is the graph of a parabola, if we start at a point above 120, then the first point that we hit 120 feet is on its way down. Note that 144 is the y-intercept so we start from there, we can only stay in quadrant one since you cannot go back in time. Plugging in 120 as y, you get 16t^2 - 28t = 24. 

4t^2 - 7t = 6

Approximately 2.38 seconds

Basically, we have a lattice grid. There total number of ways to get from the Origin to (4, 6) is 10 choose 4 = 210. The total number of ways to get from (0,0) to (2, 4) is 6 choose 2, and from (2, 4) to (4, 6) is 4 choose 2. 6 choose 2 * 4 choose 2 = 90. Subtracting, the total number of ways marvin can reach his home is 210 - 90 = 120 ways.

Well if you want to know the geeky half angle formula for contangents, here it is:

cot theta/2 = squareroot{(1 + cos theta)/(1 - cos theta)}

ta dah :D

This is a longer grueling problem: By the zero product property, we have:

x^2 + 16x - 12 = 0    => -16 +- sqrt(256 + 48) over 2, -16 +- 4sqrt(19) over 2, -8 +/- 2sqrt(19)

3x^2 - 9x + 8 = 0      => 9 +- sqrt(81 - 4*8*3), but we see its imaginary, so only consider the above case to be real.

If multiply to greater than 0, by parity, we have:

x^2 + 16x - 12x > 0    =>   x = (-infinity, -8 - 2sqrt(19)) U (-8 + 2sqrt(19), infinity)

and (3x^2 - 9x + 8) > 0 at the same time. => x = (-infinity, infinity)

By negative parities, we have:

x^2 + 16x - 12 < 0         => x = (-8 - 2sqrt(19), -8 + 2sqrt(19))

and 3x^2 - 9x + 8 < 0    => no solutions for x

Now we have to satisfy the original inequality, so we look for the most constricting ranges, obviously (-infinity, infinity) isn't helpful, and we can see that most of our bolded revolve around -8 +/- 2sqrt(19), meaning that those two values are our values of x.

Say the sides of the right triangle were x and y. Originally, the area was xy/2

Now the area is 0.3x * 1.7y / 2 = 0.51xy/2

It decreased by 49%

-6 * 3 = -1 * 6 * 3

6 * 3 = 6 + 6 + 6 = 12 + 6 = 18

when multiplied by -1, just add the negative to the number because it's reading backwards on the number line 18 times.

-18

you are not wrong, so smart :D