All Answers 358149 Answers

By the tangent - secant theorem

XY^2 =  AX * XB

9^2  =  16 * XB

81  = 16 * XB

81 / 16  = XB

So

BA =  AX - XB 

BA  = 16 - 81/16

BA =  (256 - 81) / 16

BA =  175 / 16

3

As follows:

2n^2 + 3n + 11 - n^2 + 9n  can be written as n^2 + 12n + 11 which, in turn can be factored as (n + 1)(n + 11).

So what does this tell you, remembering that a prime number has no factors other than itself and 1?

The largest possible value of b is 13.

The graph has one intercept.

c) The total probability of at least one "Josh" is the 1 - the probability of 0 "Josh" (what we don't want). 

This is called complementary counting, and it is based on the idea that if you remove what you don't want, you are left with the probability that you succeed. 

We already know from b that the probability of not getting a single "Josh" is \(0.491\), so the probability of at least one "Josh" is \(1 - 0.491 = \color{brown}\boxed{0.509}\)

b) There are \({20 \choose 4} =4845\) ways to choose the 4 students. 

However, the 3 people named "Josh" are off-limits, so there are only 17 people to choose the delegation from. 

This makes for \({17 \choose 4} = 2380\)

Thus, the total probability is \({2380 \over 4845} \approx \color{brown}\boxed{0.491}\)

a) \({20 \choose 4} = 4845 \) ways to choose the 4 members

There are 3 ways to choose which "Josh" is in the delegation. 

There is also \({17 \choose 3} = 680\) ways to choose the remaining 3 people (The 3 people named "Josh" are off-limits)

This makes for a probability of \({680 \times 3 \over 4845} \approx {\color{brown}\boxed{0.421}}\)

The line \(x = k \) can only intersect the parabola at the vertex.

The y-coordinate of the vertex is \(-{b \over 2a} = {-3 \over 4}\)

Substituting this in for y gives us: \(x = -2(-0.75)^2 - 3(-0.75) + 5\), meaning \(k = \color{brown}\boxed{6 {1 \over 8}}\)

(1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 125, 150, 200, 240, 250, 300, 375, 400, 500, 600, 750, 1000, 1200, 1500, 2000, 3000, 6000)==40 divisors

(1, 2, 3, 5, 6, 10, 15, 30)==72 - sum of all square-free divisors of 6,000

Note: You may drop 1 from the list since it is its own square!

10 balloons because 5 balloons are 1/8 of a heart and 1/4 is double of 1/8 because 1/4=2/8 and 1/8+1/8 is 2/8 so you just have to double 5 and your answer is 10 balloons!

thanks Builderboi.

That makes some sense.  

I suppose it could be an interesting assignement.

Something heavier than paper, maybe some reasonably think cardboard would give less error. 

A==Aaron,    B=Bobby

0.25B = 0.40A, 

B =A + 24, solve for A, B

A==40 beads

B ==64 beads

Hi asinus,

That was good thinking.

But I graphed your equation using Desmos and there was a solution

$\mathrm{lol \ your \ face}$

16

......

By the conditions of the problem, yes.  It says that four digits were used in the four digit passcode. 

For people out there who know the Art of Problem Solving code of conduct, it is strictly forbidden to share art of problem solving problems outisde like the one shown here. One way to notice if it is from Art of problem solving is to drag the image to another tap and see the if the link say something like: 

latex.artofproblemsolving.com..... and a string of symbols and digits. 

The link to this picture is:

https://latex.artofproblemsolving.com/8/1/9/81926fb69bf88a37491fee8cd74324715bc6bdc4.png

We have the equation: \(x + x + 1 + x + 2 + x + \cdot \cdot \cdot+ x+6 = 14(x+6)\)

Simplify to: \(7x + 1 + 2 +3 + \cdot \cdot \cdot + 6 = 14x + 84\)

Furthur simplify the left-hand side: \(7x + 21 = 14x+84\)

Subtract 7x from both sides: \(21 = 7x + 84\)

Subtract 84 from both sides: \(7x = -63\)

Divide by 7: \(x = \color{brown}\boxed{-9}\)