hectictar

Let    AB  =  w

and   BE  =  x

and   EC  =  y

The area of rectangle ABCD   =   ( length )( width )   =   15w

By the Pythagorean theorem, we can make these two equations:

w² + y²  =  14²

w² + x²  =  13²

Since ABCD is a rectangle, BC must be the same length as AD, and so we can make this third equation:

x + y  =  15

Now we can begin solving this system of three equations. Here's one way to do that:

x + y  =  15

y  =  15 - x

w² + x²  =  13²

w²  =  13² - x²

w²  =  169 - x²

w² + y²  =  14²

(169 - x²) + (15 - x)²  =  14²

(169 - x²) + (15 - x)²  =  196

169 - x² + (15 - x)(15 - x)  =  196

169 - x² + 225 - 30x + x²  =  196

-x² + x² - 30x  =  196 - 169 - 225

-30x  =  -198

x  =  6.6

w²  =  169 - x²

w²  =  169 - (6.6)²

w²  =  125.44

w  =  11.2

We don't care what  y  is, we just needed to know what  w  is.

Area of rectangle ABCD   =   15w   =   15(11.2)   =   168   sq units

The smaller end of the range of possible values of   y - x   =   -5 - 2   =   -7

The bigger end of the range of possible value of     y - x    =   5 - -2   =   7

y - x  can be anything between  -7  and  7  by starting at one end and increasing or decreasing either  y  or  x

-7  <  y - x  <  7

6^x  =  5           Raise both sides of this equation to the power of 3

( 6^x )³   =   5³           Simplify the left side using the rule  ( x^a )^b  =   x^ab   and simplify the right side.

6^3x   =  125           Multiply both sides of the equation by  6²

6^3x  ·  6²  =  125 · 6²           Simplify the left side using the rule  x^a · x^b  =  x^a+b   and simplify the right side.

6^3x+2  =  4500

false!

If you list all the possibilite outcomes

H H H

H H T

H T H

H T T

T H H

T H T

T T H
T T T

There are 3 cases where exactly one head appears, but there is only 1 case where no heads appear.

So it is more likely to get exactly one head than to get no heads.

Because they form a straight line,

m∠GFC + m∠GFB   =   180°

m∠GFC   =   180° – m∠GFB

m∠GFC   =   180° – 140°

m∠GFC   =   40°

Because the sum of interior angles in  △CGF  is 180°,

m∠GFC + m∠GCF + m∠CGF   =   180°

40°  +  59° + 3y°   =   180°

3y°   =   81°

y   =   27

Because the sum of interior angles in  △CGE  is 180°,

m∠GEC + m∠GCE + m∠CGE   =   180°

x°  +  59°  +  2y°   =   180°

x°  +  59°  +  2(27)°   =   180°

x°  +  59°  +  54°   =   180°

x°   =   180°  –  59°  –  54°

x°   =   67°

Let  b  be the base. Then convert all the numbers in the equation to base ten to get:

1b + 1   +   1   =   1b² + 0b + 0

b + 2   =   b²

0   =   b² - b - 2

0   =   (b + 1)(b - 2)

b  =  -1     or     b  =  2

Reject the negative solution, so  b = 2

We can check this answer by counting in base 2:

0

1

10

11            ←   Adding  1  more to 11

100          ←   gives us  100

101

110

111

1000

.

.

.

Glad it helped!! I learned it from one of my professors!

Also, Hi Melody!

Maybe Guest is referring to this:

\(\sum\limits_{k=1}^{n}k\ =\ \frac{n(n+1)}{2}\)

in which case, the following might help:

Here is an image to maybe help see this:

sum of orange angles  =  sum of green angles - sum of purple angles

sum of orange angles  =  4 * 180° - 360°

sum of orange angles  =  360°

(I apologize if you are colorblind!!)