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Of the three sides, the hypotenuse is the longest one.    Now $1 < \sqrt3 < 2$ so ... 

First find the number of ways to pick 2 boxes out of 5 distinct boxes where order doesn't matter.   Call this number $A$.  Next, find the number of ways to seat 3 people in a circle where two seatings are considered the same if one seating can be rotated or flipped in 3D into another.   Call this number $B$.  The answer is $A\cdot B$.

Values $\alpha, \beta$ are the zeroes of the quadratic polynomial $(x-\alpha)(x-\beta)$ if and only if they are also the roots of the quadratic equation $x^2-(\alpha+\beta)x +\alpha\beta=0$.

Diagram is missing......

Correct, tysm!

By assumption $a \ge4$ and $b\ge4$, and the assumption $32_a = 13_b$ gives the constraint $3a+2=b+3$, that is, $a = (b+1)/3$.   Solving $a=(b+1)/3 \ge 4$ gives $b\ge11$.  When $b=11$ we have $a=(11+1)/3=4$.    So the smallest possible sum $a+b=4+11=15$.

that's incorrect... sorry...

2(x)^3-2n 2(1/x)^2n = 4(1/x)^n+3 

  2(x)^(3-2n)  * 2(x)^-2n  = 4(x)^-(n+3)     **multiply on the left side to make 4(x)^m+n

 4(x)^(3-2n-2n)     =  4(x)^-(n+3)    

3-4n = -n-3

-3n = -6

I think, you can see the whole picture now.

See original post.......   ~EP

Multiply both sides of the equation by r+4  to get

r^2 -3r-17 = 2r^2 +15r + 28      simplify

r^2 +18r + 45 = 0     

(r+15)(r+3) = 0         You should be able to finish......  

SUm of the interior angles of the 10 sided polygon     (n-2)* 180 = 1440 degrees

1440/10 = 144 degrees each angle     the external angle is  180 - 144 = 36

alpha goes from 4 to 8    or   4 external angles        4 * 36 = 144   degrees

Here area couple of pictures:

Not positive of this:

5 C 2 =  10 ways for the two balls to match their boxes

    the remaining 3 boxes can only have balls distributed to them in TWO ways (Out of a possible 6) where they do not match their boxes

10 * 2 = 20 ways for exactly two to match

Assume the base area of the cylinder is $\pi r^2$. Thus, the base area of the cone is $\pi (2r)^2 = \pi 4r^2$.

It follows that the volume of the cone is $4$ times the area of the cylinder. 

However, the area of the volume of a cone is $\pi r^2 \frac{h}{3}$, notice the $\frac{1}{3}$. Can you solve the problem from here? 

Oooh thank you! Makes sense.

Let's see...

The first digit of a three-digit number obviously can't be $0$. 

Let's count the number of $0$ in the numbers $100 - 199$, as then we could just multiply by $9$ to get our result. 

For the case $101$, we have the numbers from $101 - 109$, which means there are $9$ numbers. 

For the case $110$, we can change the middle digit from $1-9$, (Eg: 110, 120, 130... 190), meaning there are also $9$ numbers. 

Thus, from $100-199$, there are $18$ numbers with exactly $1$ zero. Can you solve the problem from here? 

The last line should be \[ n = \sum_{0\le i\le k} d_i\cdot (100)^i \equiv \cdots \] instead.

Time to work =  distance / rate = d/50

Time from work =  d/75

total distance = 2d

total distance / total time =  average rate

2d / (d/50 + d/75) =   2 / ( 1/50 + 1/75) = 2 / (3/150 + 2/150) =  2/ (5/150) = 300/5 = 60 m/hr     ( as Chris found via different method !)

Parabola with vertex  -3,-2

Vertex form     y = a (x+3)^2 - 2        choose point (-1,0)    when x = -1    y = 0   shows a = 1/2

y = 1/2 (x+3)^2 -2    expand

y = 1/2 x^2 +3x + 2.5                    1/2   x 3  x 2.5 = 3.75      ( as EL found via a different method !)

LOL!!!!

Avg speed  = 

2* product of speeds  /  sum of speeds    =

(2 * 50 * 75)   /( 50 + 75)  =

7500  / 125  =

60  mph           

[   NOT      (50 + 75)  / 2  =  125 / 2  =     62.5  mph  as many think  !!!    ]

Since the graph passes through (-3, -2), (-1, 0), and (-5, 0), we have the equations \(9a-3b+c = -2, a-b+c = 0,\) and \(25a-5b+c= 0\).  Solving, we have a = 1/2, b = 3, c = 5/2, which means that abc = 15/4

Definitely NOT  a square  !!!

Easiest  way to  calculate   [ABCD     :]

Find the area of rectangle EHFB

From this  subtract  areas of

Right triangles   BEC , CFD , AGD  and Trapezoid  AGHB

ok thanks

Good to know in my everyday life .....  haha        Thanx !   ~ EP