geno3141

If the difference between any two consecutive terms is the same, then it will be arithmetic.

-173 - 27  =  -200

.373 - -173  =  -200

.573 - -373  =  -200

Since these are all the same, this sequence is arithmetic.

x + y + z  =  180

      y + z  =  4x

            z  =  12 + y

Combining the first and second equations:  x + 4x  =  180     --->     5x  =  180     --->     x  =  36

Replacing this value into the second equation:  y + z  =  4(36)     --->     y + z  = 144

Combining this with the third equation:

                y + z  =  144

                      z  =  12 + y

     --->     y + 12 + y  =  144

                            2y  =  132

                              y  =  61

--->     z  =  12 + y  =  12 + 61  =  73

But, this is impossible!  Where did I make my mistake?

For the first problem:  slope = -1     y-intercept = -2     shaded below     dashed line

                        y  <  -1x - 2

For the second problem:  slope = -1/3     y-intercept = -8/3     shaded below     solid line

                       y  <=  (-1/3)x - 8/3

Can you get these inequalities into the correct form?

Starting with the number at A, square the number, if the answer is less than 81, check that answer.

Work your way down the list.

Remember that, if you square a negative number, the answer is positive.

x⁸ - y⁸  =  (x⁴ - y⁴)(x⁴ + y⁴)

             =  (x² - y²)(x² + y²)(x⁴ + y⁴)

             =  (x - y)(x + y)(x² + y²)(x⁴ + y⁴)

This is factored as the product of four polynomials but is not the product of four polynomials

of degree 1 because that is impossible.

It is possible to further factor my result if you are allowed to use imaginary factors.

I would translate each of the percentages into a common fraction:

          33 1/3%  =  1/3

          66 2/3%  =  2/3

          11 1/9%  =  1/9

          50%        =  1/2

and then multiply the common fractions together:

          for example:  33 1/3%  x  66 2/3%  =  1/3 x 2/3  =  2/9

and then translate that common fraction back into a percentage:

          for example:  2/9  =  22 2/9%

5^1/5 x 25^1/25 x 125^1/125 x 625^1/625 x ...

=  5^1/5 x 5^2/25 x 5^3/125 x 5^4/625 x ...

=  5^{[ 1/5 + 2/25 + 3/125 + 4/625 + ...}

Looking at only the exponent:

   1/5 + 2/25 + 3/125 + 4/625 + ...

=  1/5 + 2/5² + 3/5³ + 4/5⁴ + ...

=  1/5 + (1/5² + 1/5²) + (1/5³ + 1/5³ + 1/5³) + (1/5⁴ + 1/5⁴ + 1/5⁴ + 1/5⁴) + ...

=  1/5 + 1/5²              + 1/5³                          + 1/5⁴                  + ...               = (1/5) / (1 - 1/5)  =  1/4

                      + 1/5²               + 1/5³                           + 1/5⁴      + ...               = (1/5² / (1 - 1/5²)  =  1/20

                                                          + 1/5³                         + 1/5⁴  + ...        = (1/5³ / (1 - 1/5³)  =  1/100

                ....

Summing:  1/4 + 1/20 + 1/100 + ...  =  (1/4) / (1 - 1/5)  =  5/16

So the answer is:  5^5/16     

1)  angle(R) + angle(P) + angle(Q)  =  180^o

     --->         angle(R) + 107^o + 42^o  =  180^o  

     --->                               angle(R)  =  31^o  

     p / sin(P)  =  r / sin(R)     --->     p / sin(107^o)  =  49 / sin(31^o)  

                                            --->     p  =  49 x sin(107^o) / sin(31^o)  

                                            --->     p  =  ......

The other two practice problems are solved in the same manner.

Given right triangle(ABC) with right angle(ABC) and altitude(BD).

Since AD = 4  and  BD = 2,  use the formula:  AD/BD  =  BD/DC

--->     4/2  =  2/DC

and finish.

speed is distance divided by time:  speed  =  1254 km / 3 hr  =  418 km/hr