MaxWong

Exact same question answered here: https://web2.0calc.com/questions/floor-function_20#r1

Please try to search your question before posting here.

Exact same question answered here: https://web2.0calc.com/questions/floor-function_20#r1

Please try to search your question before posting here.

We reflect (0, 5) across the line x = 5 to get the other point as (10, 5).

Number of ways to pick out 3 green marbles = $\binom{8}3$

Number of ways to pick out 3 marbles = $\binom{6 + 5 + 8}3 = \binom{19}3$.

Probability = $\dfrac{\binom83}{\binom{19}3} = \dfrac{56}{969}$.

Now you can round that to the nearest 10th of a percent, which is just the nearest 0.001.

Interesting fact: The probability is quite close to $\dfrac{1}{10\sqrt 3}$.

We count each possibility because there are not many.

2 * 3 = 6

2 * 4 = 8

2 * 5 = 10

3 * 4 = 12

3 * 5 = 15

4 * 5 = 20

2 * 3 * 4 = 24

2 * 3 * 5 = 30

2 * 4 * 5 = 40

3 * 4 * 5 = 60

2 * 3 * 4 * 5 = 120

There are 11 such integers in total. Just be careful not to miss some of the cases.

AB is perpendicular to the line y = 1/2 x + 2. So the slope of AB is -2.

Equation of AB is just y - 1 = -2(x - 3), which is 2x + y - 7 = 0.

The intersection of 2x + y - 7 = 0 and y = 1/2 x + 2 can be found via solving the system of linear equations $\begin{cases}2x + y - 7 = 0\\y =\dfrac12x + 2\end{cases}$.

The intersection point is (2, 3) and that's the midpoint of AB. Can you continue from here?

I think you forgot to post the picture with your question.

What is 36^t? That is just $(6^2)^t = 6^{2t}$

So the equation is equivalent to this one: $6^{3t - 1} = 6^{2t}$

Comparing the exponents gives a linear equation. I believe you can continue from there.

Please refer to this post: https://web2.0calc.com/questions/help-system_5

It is the exact same as this one.

Since the divisor is cubic, the remainder is at most quadratic, so we may let ax^2 + bx + c be the remainder.

By "remainder theorem", we have $\begin{cases}c = 4\\a - b + c = 5\\4a + 2b + c = 10\end{cases}$. (The details are similar to this solution here: https://web2.0calc.com/questions/polynomial_92348#r1)

The value of c is already known. You can solve for a, b here to know the coefficients of the remainder. After you get the values of all of a, b, c, the remainder will be known.