geno3141

To finish:

First, eliminate C from the last two equations:

-6A + 2B + C  =  0     --->   x 5   --->   -30A + 10B + 5C  =  0

9A - 15B - 5C  =  1   --->                        9A - 15B  - 5C  =  1

(add down)                                          -11A  - 5B            =  1

Combining this result with the first equation:

     A +  B  =  0   --->   x 11   --->   11A + 11B  =  0

-11A - 5B  =  1   --->                    -11A  -   5B  =  1

(add down)                                               6B  =  1   --->   B = 1/6

A + B  =  0   ---->   A + (1/6)  =  0   --->   A  =  -1/6

-6A + 2B + C  =  0   --->   -6(-1/6) + 2(1/6) + C  =  0   --->   1 + 1/3 + C  =  0   --->   C  =  -4/3

In the original post, there was this equation:  

   x² + 2xh + h² + y² + 2yh + h²  =  x² + h² + 2ab + y² + h²

Subtract x², h², y², and h² from both sides and you are left with:  2xh + 2yn  =  2ab

Since there are 1000 m in a km, there are 1000 x 26  =  26 000 m in 26 km

Since there are 60 sec in a min, there are 60 x 15  =  900 sec in 15 min

Since there are 60 x 60 = 3600 sec in an hour, there are 3600 x 3  =  10 800 sec in 3 hr.

There will be  900 sec + 10 800 sec  =  11 700 sec in 3 hr 15 min.

average speed  =  26 000 m  /  11 700 sec  =  2.22  meters per second

x + ay + 1  =  0   --->   ay  =  -x - 1   --->   y  =  (-1/a)·x - (-1/a)   --->   slope  =  -1/a

2x - by + 1  =  0   --->   -by  =   -2x - 1   --->   y  =  (2/b)·x + (1/b)   --->   slope  =  2/b

Since the above two lines are perpendicular, their slopes are negative reciprocals

   --->   -1/a  =  -b/2   --->   a  =  2/b   

x - (b - 3)·y - 1  =  0   --->   -(b - 3)·y  =  -x + 1   --->   (b - 3)·y  =  x - 1   --->   y  =  ( 1/(b - 3) )·x - 1/(b - 3)

   --->   slope  =  1/(b - 3)

Since this line is parallel to the first line, they have equal slopes:  -1/a  =  1/(b - 3)   --->   3 - b  =  a

Combining   a  =  2/b   with   3 - b  =  a   --->    3 - b  =  2/b   --->   3b - b²  =  2   --->   b² - 3b + 2  =  0

   --->   (b - 2)(b - 1)  =  0   --->   either   b = 2   or   b = 1

If  b = 2   --->   a  =  2/b   --->   a  =  2/2  =  1

If  b = 1   --->   a  =  2/b   --->   a  =  2/1  =  2

tan(40^o)  =  CD / AD     --->     tan(40^o)  =  CD / 10     --->     CD  =  10 · tan(40^o)

tan(30^o)  =  BD / AD     --->     tan(30^o)  =  BD / 10     --->     BD  =  10 · tan(30^o)

Use a calculator to find the lengths of CD and BD and then add those two lengths together.

1 mile  =  5280 feet

A quarter mile has  5280 feet / 4  =  1320 feet

1 foot has 12 inches.

1320 feet  =  1320 feet x 12 inches per foot  =  15840 inches

1 meter  =  39.37 inches

400 meters  =  400 meterx x 39.37 inches per meter  =  15748 inches

You can finish it from here ...

Let  O  be the center of the circle.

Let  A  be the center of the smaller left-hand circle.

Let  B  be the center of the smaller top circle.

Let  C  be the center of the smaller right-hand circle.

Let  D  be the center of the smaller bottom circle.

ABCD is a square with each side = 2.

Therefore, the area of this square is 4.

Each of the smaller circle has an area of pi·1²  =  pi

However, one-fourth of each circle is contained in the center square, so each circle contributes

another ¾·pi of area to the white portion.

Total white area  =  4 + 4(¾·pi)  =  4 + 3·pi

The distance from B to D is the diagonal of the central square. Since each side of the central square

is 2, the diagonal is  2·sqrt(2).

So, the diameter of the outside circle is  2 + 2·sqrt(2)  and its radius is 1 + sqrt(2).

The area of the outside circle is:  pi·( 1 + sqrt(2) )²,  making the area of the shaded section:

     pi·( 1 + sqrt(2) )²  -  (4 + 3·pi)

     pi

              4^x - 4^x-1  =  24

      (2²)^x - (2²)^x-1  =  24                             rewrite 4 as 2²   

           2^2x - 2^2x-2  =  24                             simplify

          2^2x - 2^2x/2²  =  24                            rewrtie 2^-2 as 1/2²

    4[ 2^2x - 2^2x/2² ]  =  4[ 24 ]                      multiply both sides by 4

           4·2^2x - 2^2x  =  96                            simplify

        4·2^2x - 1·2^2x  =  96                            rewrite

            (4 - 1)·2^2x  =  96                            factor

                   3·2^2x  =  96                            simplify

                      2^2x  =  32                            divide both sides by 3

                      2^2x  =  2⁵                             rewrite

                       2x  =  5                               rewrite

                         x  =  2.5                            divide

(2x)^x  =  (2 · 2.5)^2,5  =  5^2.5  =  5² · 5^0.5  =  25·sqrt(5)

If  n = 1,  (7n + 1)/(3n + 5)  =  1 

If  n = 9,  (7n + 1)/(3n + 5)  =  2.

The quotient  (7n + 1)/(3n + 5)  can never reach 3 because, as n approaches infinity,

the  lim (7n + 1)/(3n + 5)  = 7/3.

There are only two possibilities. 

Jack has 2 more chocolates than Jill.

They can't have very many; so, start low:

If Jack has 3 and Jill has 1, then, when Jill has given one chocolate to Jack, Jack will have 4 and she won't have any, so Jack's number won't be twice as many as Jill.

Try 4 and 2, then 5 and 3, then 6 and 4, then 7 and 5 --

At 7 and 5, if Jack gives one to Jill, they both will have 6 and if Jill gives 1 to Jack, he'll have 8 and she'll have 4 (and Jack will have twice as many as Jill).

Jill has 5.