All Answers 358181 Answers

We can use the Law of Sines to answer this

sin R  / 30   = sin 96° / 54        multiply both sides by  30

sin R  = 30 * sin 96° / 54

To find  "R"....we  need to use the  sine inverse   ( arcsin )

arcsin [ 30 * sin (96°) / 54  ]  =  "R"   ≈ 33.54°

It seems confusing at first, but after you explaining, I understand how you got the final answer, Thanks!  

x^3=125

take cube root on both sides

So,  x=5

48.....We can use the tangent inverse to find this

atan (5.67)  =  79.997°  ≈   80°

49.

sin^2  θ                 1   - cos^2  θ            ( 1 - cos  θ) ( 1 + cos  θ)                   1  +  cos  θ

_________   =      __________    =      ____________________   =  

1  - cos  θ               1  - cos  θ                (1  -  cos  θ)

Cubing the  2  and putting it back under the root   would result  in  2^3  = 8

Cubing  the a^3  and putting it back under the root would result  in  (a^3)^3  =  a^9

Cubing  the b^2  and putting it back under the root  would result in  (b^2)^3  = b^6

So....we  must have had

³√  [ 8 * 12 * a^9 * b^6 * b  ] = 

³√ [ 96 * a^9  * b^7 ] 

Does that make some sense, LF  ????

4.

I appreciate you working all that out!! 

Remember that 

Area  = Width  * Length   ......so we have this.....

[8x^3 + 10x^2 - x  - 2]   =  [2x  + 1]   *   Length     ....so...dividing both sides by   2x  + 1  , we have

[ 8x^3  + 10x^2  - x  - 2   ] / [2x + 1  ]  =  Length 

We can find the length with some  polynomial long  division  (yuck !!!)

               4 x^2  + 3x   - 2

2x + 1  [  8x^3  + 10x^2   -  x   - 2  ] 

               8x^3  +  4x^2

            _______________________

                           6x^2  -    x 

                           6x^2  + 3x

                          ______________

                                      -4x   -  2

                                      -4x   -  2

                                     _________

                                                  0

The Length is the polynomial shown in red   ⇒  "C"

3.

Thank you!! I get it!! 

y  = 2 ( x + 1/4)^2  -  25/8

Remember.....the  y intercept point has an x value  = 0.....so...let   x  = 0....and we have

y  =2 (0 + 1/4)^2  - 25/8

y  = 2 (1/4)^2  - 25/8

y  = 2 (1/16) - 25/8

y  = 1/8  - 25/8

y   = -24/8    =   -3   ....just as you found, LF   !!!!

After graphing, I found it was -3, but how do you find that besides graphing? 

2. 

To find the number of people who use the fitness suite only...we can  take 57 - 36  = 21

To find the number of people who only use the swimming pool, we have 49 - 36 = 13

To  find those that use neither, we have :   100 - 36  - 21 - 13  = 30

Here's the diagram :

Oh!!! I understand now! It means the Y values, from one to another!! Thanks! 

"Range" refers  to  the y values covered by the graph....these are   [ -3, 6 ]

The y values  - not "points" -  in this range  are  ( -3, 0, 1, 5 )

1.

We have  a total of 10 students  who have either  a brother or a sister....

To find the number that goes  in the middle, x,  we can solve this

B  +  S   -  x  = 10

9 + 7  - x  =  10

16  - x  = 10

16  - 10  = x 

6  = x

Here's the diagram :

Thank you!!

Well Thanks!!! I saw the first one, but wasn't sure! 

Let's look at the first few terms

-4 +  2  - 1 +  1/2.......

The common ratio, r,  =  -1/2   .....since  l r  l  < 1....this will converge

And the first term is  -4

So...the sum  is given by

[ first term  ]  /  [ 1  - r ]  =

-4  /  [ 1 - (-1/2) ] =

-4  /  [ 3/2]   =

-8 / 3

We have the equation

y  = Asin (Bx)

The amplitude  =   A   = 4   

Finding B   is a little tricky.....we want to  solve this  :

2pi / Period  = B

2pi  / [ 3pi / 2 ]  = B

4pi / 3pi  = B

(4/3)  = B

So...the equation becomes

y  = 4sin (4/3 x)

Here's the graph for one period   :  https://www.desmos.com/calculator/ncw51ksw5x

Here's the graph  :

https://www.desmos.com/calculator/7aqynw02yn

52

1/x  + 1  / (x + 2)   =   9/40    get a common denominator on the left

[ (x + 2 )  +  x ]  /  [ x (x + 2) ] =    9/40        cross-multiply

40 [ x + 2 + x   ] =  9 [ x(x + 2) ]      simplify

40  [ 2x + 2]   =  9 (x^2 + 2x )

80x + 80  = 9x^2  + 18x

9x^2  + 18x   =  80x  + 80        subtract  80x, 80  from  both sides

9x^2  - 62x  - 80  = 0       factor

(9x + 10) (x - 8)  = 0     set each factor to 0  and solve for x

9x  + 10  = 0            x - 8  = 0

9x  = -10                   x  = 8

x  = -10/9

Since we must have interger answers.....x  = 8    and  x + 2  =  8 + 2  = 10

50.  2sinθcosθ + √2 cos θ   = 0          factor out cosθ

cosθ (2sin θ + √2 )  = 0

So...either  cosθ  =  0      and this happpens at  pi/2  and  3pi/2     ......or....

2sin θ + √2   = 0       subtract √2  from both sides

2 sinθ  =  -√2    divide both sides by  2

sinθ  =   -√2 / 2     and this happens a 5pi/4    and at 7pi/4

So...the answers are    pi/2 , 3pi/2./  5pi/4 and  7pi/4   ⇒   the third answer

51.     √(7x) [ √x - 7√7  ] =

 √(7) x- 49√x

A. AB  is the hypotenuse of a 30-60-90 right triangle....it will be twice as long as the  side opposite the 30°  angle  = CB

So....AB   = 2 CB  =  2 (271)  =  542 ft

B. We need to find  AC...it will  √3  times as long as BC   = 271√3 ft

So...using the Law of Cosines....we have

AD^2  = AC^2  + DC^2  - 2 (AC) (DC) cos ( 74°)

AD  =  √ [  (271√3)^2  + (772)^2  - 2 (271√3)(772)cos (74°) ]  =  785.2 ft

C.  We can find angle E  as follows :

DC^2  = DE^2  + CE^2  - 2(DE)(CE)cos (E)

772^2   = 543^2 + 561^2  - 2(543)(561) cos E

[ 772^2  - 543^2 - 561^2 ]  /  [ -2(543)(561) ]   = cos E

Take the inverse cosne to find E

arccos  ( [ 772^2  - 543^2 - 561^2 ]  /  [ -2(543)(561) ] )  = E  = 88.7°