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I'm assuming that you're saying that one side of the quadrilateral is the same as the circle's diameter, and we have two adjacent sides to this that are 8 and 12m, respectively. Then the circle's radius = 10 m

To find the remaining side, we need to first find the central angles intercepting the 8 and 12 m sides.....

We can use the Law of Cosines in both cases....for the 12m side, we have

12^2 = 10^2 + 10^2 - 2(10)(10)cosΘ   ......simplifying, we have

cosΘ= 7/25  = cos^-1(7/25) = about 73.739795291688°

For the 8m side, we have

8^2 = 10^2 + 10^2 - 2(10(10)cosΘ   ...simplifying again, we have

cosΘ = 17/25 = cos^-1(17/25) = about 47.156356956404°

So, since the diameter would span 180° of arc, the remaing side must span (180 - 73.739795291688 - 47.156356956404)° = 59.103847751908° of arc

So...the remaining side... (s).... is given by

s^2  = 10^2 + 10^2 - 2(10)(10)cos(59.103847751908)

s = √[10^2 + 10^2 - 2(10)(10)cos(59.103847751908)] = about 9.86 m

This makes sense....the greater side intercepts the greatest arc, the next greatest side intercepts the next greatest arc, etc.

The product of $$\sqrt{a}$$ and $$a^{\frac{1}{2}}$$ is $$a^{\frac{1}{2}}\times a^{\frac{1}{2}}=a^{\frac{1}{2}+\frac{1}{2}}=a^1=a$$

.

The general rule here is $$a^n\times a^k=a^{n+k}$$

.

As x gets very large ln( (x+1)/(x-2) ) → ln( x/x ) → ln(1) = 0

and 1/√(x² - 4) → 1/√x² → 1/x → 0

.

AΔ= (base x height) / 2

Do the oposite operation

6=1/3 (what was left)

X=6*3 (total before you eat 2/3)

X=6*3=18

-----|-----

18=1/3

Y=18*3 (total of chocolates before you eat)

Y=18*3=54//

it is $${{log}}_{{\mathtt{24}}}{\left({\mathtt{50}}\right)}$$ so it is 1.230949

what properties does it have?

x^2 - 9  =

(x + 3) (x - 3)

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{9}} = {{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{{\mathtt{3}}}^{{\mathtt{2}}}$$ but it is quite the simplest form

i didn't calculate it but if it is 0/0 and the derivate of  is 0 then the expression's derivate should be derivate of the derivate of the divident divided by derivent of the divisor

so we derivate both to solve the equotation

i think

the area of a rectangle is a*b so it is $${\mathtt{17.62}}{\mathtt{\,\times\,}}{\mathtt{13.7}} = {\mathtt{241.394}}$$

21 duh

The area of the whole trapezoid is(1/2)(18 + 12)(6)= (15)(6) =   90m^2

Let the upper base = 12 and let the lower base = 18

And since a line divides the area in a ratio of 2:3, let the area of the upper trapezoid = 36 and let the area of the lower trapezoid = 54

Let 12 + 2x be the length of the dividing line and let y be the height of the lower trapezoid...so we have

(1/2)(12 + 2x + 18)(y) = 54

(30 + 2x) y = 108

2(15 + x)y = 108

(15 +x)y = 54

y = 54/(15 + x)

And the area of the upper trapezoid can be represented by

(1/2)(12 + 12 + 2x)(6-y) = 36

(24 + 2x)(6-y) = 72  .....  factor out a "2"

(12 + x)(6-y) = 36

6 - y = 36/(12 + x)

y = 6 - 36/(12 + x)

And equating "y's"     ....we have

54/(15 + x) = 6 - 36/(12 + x)   and soving for x, we have

x = 3(√6 - 2) = about 1.3484692283495343m....so 2x = about 2.6969384566990686m

So...the length of the dividing line is about (12 + 2x)m = 14.6969384566990686m

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Note that this problem would probably have a different answer if we kept the orientation of the bases the same and considered the upper trapezoid to contain 3/5ths of the area and the lower one to contain 2/5ths of the area. I'll let the adventurous types work through that one....!!!!

Is your question "What does the square root of 2 mean?" ? If so then the square root of 2 is the number that satisfies number * number = 2. Just like the square root of 9 is 3 because 3 * 3 = 9,  the square root of 16 is 4 because 4 * 4 = 16, the square root of 25 is 5 because 5 * 5 = 25,.....

   What is the value of square root 2? Its value is not a whole number but a decimal number. Your calaculator will give as 1.414...

What an informative answer!

i somehow found your translating text a bit funny!it seriously might have been a pain doing all that!

i really liked you answer!

thanks anonymous!

you surely deserve a point from me!

@@ End of Day Wrap        Sun 28 /12/14       Sydney,   Australia      Time 1:25am (really Mon)      ♪ ♫ 

hi everyone, 

There were were a few more questions today and some were quite good ones.

Our answerers were gino3141, CPhill, roasala, school, tyvlad012 and Alan.  Thanks all.

Interest posts:

1) Here is one for middle level trig students to have a go at - I have not actually finished it.  

Circumference of a circle = 2πr

So in order to find the circumference of this circle, we need to figure out what the radius is, which we can do by setting 49π equal to πr²

49π = πr²

Like terms cancel, leaving us with:
49 = r²

Solving for r, we get:
r = -7 or r = 7

Since it doesn't make sense to have a negative radius, we can ignore r= -7 as a potential answer, so now, we can go ahead and figure out the circumference, like so:
Circumference = 2π(7)

Final Answer:
Circumference = 14π

okay

when you cut the peice of wire let the peices be x and 50-x  metes long     (that works because 50-x+x=50)

so

one of the squares will have side lengths of x/4 metres and the other will have sides of (50-x)/4 metres

So

The areas if the squares will be       $$\left(\frac{x}{4}\right)^2+\left(\frac{50-x}{4}\right)^2$$

so

$$\begin{array}{rll}
\left(\frac{x}{4}\right)^2+\left(\frac{50-x}{4}\right)^2&=&100\\\\
\frac{x^2}{16}+\frac{2500-100x+x^2}{16}&=&100\\\\
\frac{2x^2-100x+2500}{16}&=&100\\\\
2x^2-100x+2500&=&1600\\\\
2x^2-100x+900&=&0\\\\
x^2-50x+450&=&0\\\\
x&=&\frac{50\pm\sqrt{2500-1800}}{2}\\\\
x&=&\frac{50\pm\sqrt{700}}{2}\\\\
x&=&\frac{50\pm10\sqrt{7}}{2}\\\\
x&=&25\pm5\sqrt{7}\\\\
\end{array}$$

now one of the peices will be   $$25+5\sqrt7$$   and the other will be  $$25-5\sqrt7$$

the difference between these will be      $$25+5\sqrt7-(25-5\sqrt7)=10\sqrt7\;m$$

the difference between the sides will be a quarter of this =  $$2.5\sqrt7\;\;metres$$ 

Actually, it also depends on which system you use. Where I come from (Norway), the next number in the sequence is trilliard. That is because in continental Europe and Latin America, one usually use "the long scale", while in USA, the short scale is used. Whenever I read large numbers in an English text, I have to "translate" it from the short scale, back into the long scale that I use. To be honest, that can be a pain in the neck, and quiet a bit confusing:

When I read "billion" I translate it into "milliard"

When I read "trillion" I translate it into "billion"

When I read "quadrillion" I translate it into "billiard"

When I read "quintillion" I translate it into "trillion"

As you might have noticed, numbers such as "billion" means different things depending on whether the long - or the short scale is used! 

To read more about this, visit "The Long and Short Scales" Wikipedia page. 

Mon 29/12/14

The first four were all suggested by CPhill. I am always pleased when people request interest posts to be included.  Thanks Chris  

1)  Simplifying indices - A pretty cool question

the square root of 2 is an irrational number.   It can be approximated but it cannot be written exactly using numerals

youcan use the calc to estimate it if you wish.

put it in as 

sqrt(2)=   

or

2^(1/2)=

Well 

the perimeter of the semicircle is   2r+0.5(2pi*r) = 2r+pi*r

the perimeter of this particular semicircle is 14+7pi

Let the sides of the rectangle be x, x, x+1, and x+1

So

$$\\x+x+x+1+x+1=14+7\pi\\
4x+2=14+7\pi\\
4x=12+7\pi\\
x=\frac{12+7\pi}{4}$$

so the area of the rectangle is

$$\\Area=x(x+1)\\\\
=\frac{12+7\pi}{4}\left(\frac{12+7\pi}{4}+1\right)\\\\$$

and you can do that on the calc on the home page.  It will be in cm squared.

but still i have a human brain......and brain doesnt take knowledge by age!does it?nah!

area = 32 * 6    cm squared

Well going by your icon you are very young :)