CPhill

BC = √2

Angle BFC = 105°

Angle BCF = 45°

Using the Law of Sines

sin BFC / √2 =   sin BCF / BF

sin 105 / √2  = 1 / [ √2 BF ]

sin 105 =  1 /BF

sin[ 60 + 45] = 1/BF

sin 60cos45 + sin45cos60 = 1/BF

√3/2 *√2/ 2 +  √2/2 *1/2 = 1/BF

[ √6  + √2 ] / 4  = 1/BF

4 / [ √6 + √2 ]  = BF

4 [ √6 - √2] / 4  = BF

√6 - √2   =  BF

And because by ASA  triangle BAE is congruent to Triangle BCF....then BE = BF

So.....area of Triangle  BEF  =

(1/2) BE * BF sin (30)  =

(1/2) [ √6 - √2 ] ^2  (1/2)

[ 6 - 2√12 + 2 ] / 4  =

[8 - 4√3 ] / 4  =

2 - √3  =

2 - 1 √3

So

a + b + c  =    2 + 1 + 3   =   6

1. If x is a real number such that 2^(2x+3)=14, find 2^x.

2^(2x + 3)  = 14         

2^(2x) * 2^3  = 14

2^(2x) * 8  =  14       divide both sides by 8

2^(2x)  =  14/8

2^(2x) =  7/4       and we can write

(2^x)^2 =  7/4        since an exponential can never be nrgative,   take the positive root

2^x  =  √7 / 2

Thanks, guys.....!!!!

3. Three of the four vertices of a rectangle are (5,11), (16,11) and (16,-2). What is the area of the intersection of this rectangular region and the region inside the graph of the equation (x-5)^2+(y+2)^2=9? Express your answer as a common fraction in terms of pi.

The remaiining vertex of the rectangle is (5, -2)

And this is the center of the circle, as well

Here's a graph :

The region is just the area of a quarter-circle with a radius of 3  =

pi * (3)^2  / 4 =

9/4  pi units^2

Thanks to PM for spotting my earlier mistake......!!!

164x + 37y  =  1

164 = 4*37 + 16

37 =  2*16 + 5

16 = 3*5 +  1

Working backwards, we have

1 = 16 - 3(5)

1 = 16 - 3[ 37 - 2(16) ]

1 = -3(37) + 7(16)

1 = -3(37) + 7 [ 164 - 4*37 ]

1 =  -3(37) + 7(164) - 28 (37)

1 = 164(7) + 37(-31)

(x, y) = (7, -31)

Excellent, tertre......!!!!

41% = .41

So.....

41% of 28244.66 =

.41 *  28244.66 =

$11580.31

Notice that we have 3 + 4 + 5 =    12 equal parts of 120m  = 10 m each 

3 of these  =  30m

4 of these = 40m

5 of these i= 50m

A 3-4-5  triangle is a Pythagorean right triangle....so.....each side of this triangle is 10 times as much

So..We have a right triangle  with legs of 30m and 40m  and a hypotenuse of 50m

The area is     product of legs / 2  =   30 * 40 /  2    =    600 m^2

Good job, PM......!!!!!

The roots are

13 + 3√17                        13 - 3√17

_________     and          _________

     2                                       2

So

The reciprocal  sum is

      2                       2

_________  +    _________

13 + 3√17            13 - 3√17

  2 [ 13 - 3√17]  + 2 [ 13 + 2√13 ]

__________________________

        169  - 9(17)

  52              13

____  =       ___

  16               4