hectictar

sin of an angle   =  length of opposite side / length of hypotenuse

sin 60°   =   y / 8

                             Multiply both sides of the equation by  8 .

8 sin 60°  =  y

y  =  8 sin 60°

                             sin 60°  =  √3 / 2

y  =  8( √3 / 2 )

y  =  4√3

\(\frac{3g}{g^3-9g}\)

To find the non-permissible value of  g , set the denominator equal to zero.

g³ - 9g   =  0    To solve this equation for  g , we need to factor the left side.

                        Start by factoring a  g  out of both terms.

g(g² - 9)  =  0   Now we can factor  g² - 9  as a difference of squares.   g² - 9  =  (g + 3)(g - 3)

g(g + 3)(g - 3)  =  0      Set each factor equal to zero.

g  =  0     or     g + 3  =  0     or     g - 3  =  0

                          g  =  -3                 g  =  3

The non-permissible values of  g  are  0 ,  -3 , and  3

Happy Pilcrow Day!!!

Happy Anniversary!!!

You can enter this directly into the calculator on the home page.

Try typing in

ln(31)

and press enter/equals. You should get  3.4339872044851462

which rounds to  3.4340

(Since there is no right angle, there is no hypotenuse )
 

A = 40°, B = 60°, c = 20

Let  a  be the length of the side opposite angle A.

Let  b  be the length of the side opposite angle B.

Let  C  be the measure of the angle opposite side c .

The sum of the angle measures in a triangle is  180°  , so...

A + B + C  =  180°

C  =  180° - A - B

C  =  180° - 40° - 60°

C  =  80°

Using the Law of Sines...

\(\begin{array}\ \frac{b}{\sin B}&=&\frac{c}{\sin C} \\ \frac{b}{\sin 60°}&=&\frac{20}{\sin 80°} \\ \quad b&=&\frac{20\sin60°}{\sin 80°} \\ \quad {\color{purple}b}&{\color{purple}\approx}&{\color{purple}17.6} \\~\\ \frac{a}{\sin A}&=&\frac{c}{\sin C} \\ \frac{a}{\sin 40°}&=&\frac{20}{\sin 80°} \\ \quad a&=&\frac{20\sin40°}{\sin 80°} \\ \quad {\color{purple}a}&{\color{purple}\approx}&{\color{purple}13.1 } \end{array}\)

(Also...this is not the ambiguous case...the ambiguous case is SSA)

2.5           

P(x)   =   a(x + b)²(x - c)

P(x)   =   a(x + b)(x + b)(x - c)      Multiplied out.....

P(x)   =   ax³ - acx² + 2abx² - 2abcx + ab²x - ab²c     So  P(x)  does have a degree of  3 .

P(x)  has multiplicity of  2  at  (-3, 0)  and passes through  (2, 0)  so...

P(x)  =  a(x + 3)²(x - 2)

P(x)  passes through  (0, 36)  so  P(0)  =  36

P(0)  =  a(0 + 3)²(0 - 2)

36   =   a(0 + 3)²(0 - 2)

36   =   a(9)(-2)

36   =   -18a

a   =   -2

0  =  -x - 3

                      Add  x  to both sides of the equation.

x  =  -3

-3  is  ≤  1  so  -3  is a solution to  f(x) = 0

0  =  x/2 + 1

                      Subtract  1  from both sides.

-1  =  x/2

                      Multiply both sides by  2 .

-2  =  x

-2  is not  >  1  so  -2  is not a solution to  f(x)  =  0

The only value of  x  that makes  f(x) = 0  is  -3

So the sum of all values of  x  such that  f(x) = 0  is just  -3 .

Part A:

We want the equation of the line with a slope of  4  that passes through the point  (1, 2)

We know a point  (1, 2)  and we know the slope  4

So we can write the equation in point-slope form.

y - 2   =   4(x - 1)            To get this into slope-intercept form, distribute the  4 .

y - 2   =   4x - 4              Add  2  to both sides.

y  =  4x - 2

Part B:

Is the line   x + 4y - 20  =  0   perpendicular to the line  y  =  4x - 2   ?
 

The slope of the line   y  =  4x - 2   is  4.

To find the slope of the line  x + 4y - 20  =  0  , let's get the equation into slope-intercept form.

x + 4y - 20  =  0       Add  20  to both sides.

x + 4y  =  20            Subtract  x  from both sides.

4y   =   -x + 20         Divide both sides by  4 .

y   =   -\(\frac14\)x + 5          Now we can see that the slope of this line is  -\(\frac14\) .

Since   -\(\frac14\)   is the negative reciprocal of  4  , these lines are perpendicular.

Yes, the two ships are sailing perpendicular to each other.