geno3141

9. · 8.  +  6.  +  4.  +  75. · 0. 

If you place this angle on a graph, it will be in the third quadrant, 45° below the x-axis.

The sides of a  45° - 45° - 90°  triangle are  1 - 1 - √(2)

Labeling the three sides: the hypotenuse is '1', the horizontal (x-value) is '-1', and the vertical (y-value) is also '-1'.

The value of sin is the y-value divided by the hypotenuse:  sin(225°)  =  - 1 / √(2).

Rationalizing the denominator,  sin(225°)  =  - √(2 / 2).

Let us say that each side of the original block has length 'x'; then its volume is  x³.

If you make each side twice as long, each side will have length '2x'; then its volume is (2x)³  =  8x³.

So, the new cube will have 8 times the volume of the old cube.

If the old cube weighs 6 pounds, the new cube will weigh 6 x 8  =  48 pounds.

c³ + 4c > 5c²

Rearrange:  c³ - 5c²+ 4c > 0

Factor:   c(c² - 5c + 4) > 0          ----->         c(c - 4)(c - 1 ) > 0

Each section could either be positive or negative:

If all the factors are positive, the answer would be positive:  c > 0  and  c - 4 > 0  and  c - 1 > 0

           which is this:  c > 0  and  c > 4  and c > 1    which is true only when c > 4.

If two of the factors are positive, the answer will be negative, so we don't have to consider this.

If only one of the factors is positive, the answer will be negative; there are three possibilities:

          c > 0  and  c - 4 < 0  and  c - 1 < 0   --->   c > 0  and  c < 4  and  c < 1   --->   0 < c < 1

          c < 0  and c - 4 > 0  and  c - 1 < 0   --->   c < 0  and c > 4  and c < 1  (impossible)

          c < 0  and  c - 4 < 0  and  c - 1 > 0   --->  c < 0  and  c < 4  and  c > 1   (impossible)

If none of the factors is positive, the answer will be negative, so we don't have to consider this.

Answer:  Combining the first answer with the third answer:    {c > 4}  or  {0 < c < 1} 

Yes; a negative is less than a positive; that's why the answer includes the word "or"; the answer is either in the first section (less than the negative) or in the second section (more than the positive); it can't be in both.

Good luck in the next world!

Using deMoivre's formula:

(1cis60)^1991  =  (1^1991)cis(60*1991)  =  cis(119460)  =  cis(300)  =  .5 + -√(3)/2i

(1cis(-60))^1991  =  (1^1991)cis(-60*1991)  =  cis(-119460)  =  cis(-300)  =  .5 + √(3)/2i

Adding:   .5 + -√(3)/2i  +  .5 + √(3)/2i  = 1 + 0i  =  1

Consider three cases:

1) z = -1: this is impossible because the denominator would become zero.

2) z > -1  (This makes the denominator positive; so when you multiply both sides by z + 1, you do not

                     need to change the direction of the inequality.)

     (3 - z) / (z + 1) ≥ 1

     Multiply both sides by z + 1:

      3 - z ≥ 1(z + 1)

      3 - z ≥ z + 1

      2 - z ≥ z

         2  ≥ 2z

         1 ≥ z     --->  z ≤ 1 and z > -1     --->     -1 < z ≤ 1

3)  z  < -1  (This makes the denominator negative; so when you multiply both sides by z + 1, you have

                     to change the direction of the inequality.)

     (3 - z) / (z + 1) ≥ 1

     Multiply both sides by z + 1:

      3 - z ≤ 1(z + 1)

      3 - z ≤  z + 1

      2 - z ≥  z

         2  ≤  2z

         1 ≤  z     --->  z ≥ 1  (This is imposible since the assumption of this section was z  < -1.)

Conclusion:  the answer is:  -1 < z ≤ 1      <==============               

If you have a negative exponent in the numerator, write it in the denominator with a positive exponent.

Example:   x^-7  =  1 / x^7

Example:  3x^-4 / 14  =  3 / (14x^4)

Example:  3x^-2y^5z^-4  =  3y^5 / (x^2z^4)

If you have a negative exponent in the denominator, write it in the numerator with a positive exponent.

Example:  1 / x^-8  =  x^8

Example:  3x^3 / 2y^-6  =  3x^3Y^6 / 2

Example:  4x^5y^-2Z^-5 / (3x^-3Y^6z^-3)  =  4x^5x^3z^3 / (3y^2y^6z^5)  =  4x^8 / (3y^8z^2)