CPhill

The diagonal of the square must be  <  9  cm  for the square to fit

The diagonal is given  by

diagonal  =     sqrt  ( side^2  + side^2)    = sqrt  [ 5^2 + 5^2 ] = sqrt  [ 50]  ≈  7.07  cm

So..the square will fit inside the circle

Total Income  =

2300 + 640   =   2940   (1)

Total  Expenses  =

775  + 308  +  385 + 273  + 87  + 30    = 1858   (2)

Money left  =    (1)  - (2)  =   2940   - 1858    = $1082

See the following  image :

AZDY  forms a  kite with angles AZD, AYD  = 90    

And  ZAY  = 40

So...angle ZDY =    360 - 2(90)  - 40  = 140

And in triangle ZDY, because DZ  = DY....then triangle ZDY is isosceles with angle YZD  = [ 180 - 140 ] / 2  = 

40/2   = 20

And CXDZ  also forms a kite with angles DXC, DZC  =  90

And angle ZCX  =  80

So....angle ZDX  =  360  - 2(90) - 80  =  100

And in triangle ZDX, because DX  = DZ....then triangle ZDX is isosceles with angle DZX  = [ 180 - 100 ] / 2  =

80/2   =  40

And angle YZX   = angle YZD  + angle DZX  =  20 + 40    =  60

When A is reflected over the x axis to B....the new point is  ( 3, -4)

Call this the base of the triangle AB  = 8  units

And the equation  of the line on which this base lies is  x  = 3

When B is reflected over the line y  = x  the new point C has the  coordinates  ( -4, 3)

The  distance from this point to the line  x  =  3  is the height of the triangle  = 7

So....the area  of ABC   = (1/2) AB * 7   = (1/2) (8)(7)    = 28 units^2

Not great at Physics, but I'll give it a shot

vf^2  = vi^2 + 2ad

vi  = initial velocity  = 0....so we have

(50m/s)^2  =  0    + 2 (4m/s^2) * d

2500m^2 / s^2  =   8 m/s^2  * d          divide both sides by  8m/s^2

[2500m^2/ s*2]  / [ 8m /s^2]   = d   =  312.5 m  ......correct  !!!

Second one

vi  = 20m/s

vf  = 0   [ we are stopped at the end ]

a  = -5m/s^2   [ we are slowing down ]

So

0  =  [20m/s]^2  + 2(-5m/s^2) * d

0 =  400m^2 / s^2     - 10 m/s^2 * d       multiply through by  s^2

0  = 400m^2  - 10m  * d

10m * d  =  400m^2        divide both sides by 10m

d  = 400m^2  / 10m   =  40 m  = stopping distance

Two vertices of an obtuse triangle are (6,4) and (0,0). The third vertex is located on the negative branch of the x-axis. What are the coordinates of the third vertex if the area of the triangle is 30 square units?

The height of the triangle is  4

So

Area of  triangle  =  (1/2) base * height

30  = (1/2) base * 4      simplify

30  = base * 2        divide both sides by 2

15  = base

The third vertex is   ( -15,  0)

Since AB  and CD are parallel

Then angles A  and D are supplemental  ....so

A  + D   = 180

2D + D   = 180

3D  = 180

D   = 60

And  A   = 2D   = 2(60)   = 120°

We have either

y  = 2x   or   x  =  2y

Look at their graphs, here : https://www.desmos.com/calculator/t2gtfjbzrd

So....they split the plane into 4 regions

x^2 - 5x + 4  = 0

Factor  the left side as  :

(x - 1) ( x - 4)  = 0

Set both factors to 0   and solve for x and we have that   x  = 1, 4

Using dimensional analysis

22.059 micrometers  x 1 cm / 10000 micrometers x 1 in / 2.54 cm

[ cancel the units and we have ]

[22.059 / [ 10000 * 2.54] in  = 0.0008684 in   =  8.684 x 10^-4 in