CPhill

We can use this

sin 60  =  x  / 4

√3 /2  =  x  / 4     multiply both sides by 4

4√3 / 2  =  x

2√3 m =  x

We have this

m XYZ  =  (1/2) (m XSZ  - m XZ )  =  (1/2) ( 220 -  140 )  = (1/2) (80)  = 40°

I don't see any figure.....

Mass  / Volume   =  Density

So....multiply both sides by  Volume

Mass  =  Density * Volume

Mass  =  4.5g / cm^3   * 0.4 cm^3

The cm^3  cancels....and we're left with

Mass  =   4.5 g  *  0.4   =   1.8 g

10000x^2  - 64   =  0       factor as a difference of squares

(100x - 8) (100x + 8)  = 0

Set both factors to 0  and solve for x

100x  - 8   =  0                    100x  + 8  = 0              

100x  = 8                             100x  =  - 8

x  = 8/100  =  2/25                x  =  -8/100  =  -2/25

9x^2  - 8  =  -34        rearrange  as

9x^2 + 34x - 8  =  0    factor

(9x - 2) (x + 4)  =  0

Set both factors to 0  and solve for   x

9x - 2  = 0                 x  +  4  =  0

9x  =  2                      x =  - 4          

x  = 2/9

2x^2 - 4x + 7  = 0     us the quadratic formula

x  =    4 ± √ [ 4^2 - 4*2* 7 ]           4  ± √ [-40]           4  ± 2 i√ 10

         ________________   =     __________   =    _________  =

              2 (2)                                    4                            4

2  ± i√10

_______

    2

3.2x + 0.2x^2 - 5  = 0      rearrange

0.2x^2 + 3.2x - 5  =  0     multiply through by 10

2x^2 + 32x - 50  = 0        divide through by 2

x^2 + 16x - 25  =  0       complete the square on x

x^2 + 16x + 64  =  25 + 64

(x + 8)^2  =  89         take both rots

x + 8   =   ± √89

x  =   ± √89  - 8

13  Construct an altitude to Hypotenuse c  and label it CD

14  I answered this one, here :

https://web2.0calc.com/questions/help-please_34288#r2

Since Jordan's position is set.......the total possible arrangements are

3  ways to pick the runner in the first position

2  ways to oick the runner for the second position

1 way to select the rnner for the third position

So...3 * 2 * 1  =    6 different orders

First one...we can use the Law of Sines

sin R / 30  =  sin 69 / 35       multiply both sides by 30

sin R  = 30 sin (69) / 35

Use the arcsin to find R

arcsin [ 30 sin (69) / 35 ]  =  R  ≈   53.15°  =  53°

Second one.....

Note that angle F  =  180  - 87  - 40  =  53°

So...we can use the Law of Sines, again

x / sin 87  = 13 / sin 53     multiply both sides by sin 87

x  =  13 * sin 87  / sin 53     ≈ 16.25 cm   =  16 cm  (rounded)

sin (40)  = cos (3x)

These angles will sum to 90°  since they are co-functions....so...

40  + 3x  =  90      subtract  40  from both sides

3x  =   50     divide both sides by 3

x  = 50 / 3

OK...look  at the following  :

So...the total number of students  is   8 + 5 + 13   =  26