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Solve for p:
(-125 i - 5) + 5 p = 10000

Add 5 + 125 i to both sides:

5 p + ((-125 i - 5) + 125 i + 5) = 125 i + 5 + 10000

(-125 i - 5) + (125 i + 5) = 0:

5 p = 10000 + 5 + 125 i

10000 + 5 + 125 i = (10000 + 5) + 125 i = 10005 + 125 i:
5 p = 125 i + 10005

Divide both sides of 5 p = 125 i + 10005 by 5:
(5 p)/5 = (125 i + 10005)/5

p = (125i + 10005)/5

Since the sum of the angles in a triangle is  180°  and two of the angles are  90°  and  45°,

the measure of the third angle   =   180° - 90° - 45°

the measure of the third angle   =   45°

Since two of the angles in the triangle are 45°, it is an isosceles triange, and the sides opposite the 45° angles have the same length. So the unlabeled side has a length of  x .

Now we can use the Pythagorean theorem to find  x .

(leg)² + (other leg)²   =   (hypotenuse)²

x² + x²   =   10²

x² + x²   =   100

                               Combine like terms.

2x²   =   100

                               Divide both sides by  2 .

x²   =   50

                               Take the positive square root of both sides.

x   =   √50

                               Simplify the radical.

x   =   √[5 * 5 * 2]

x   =   5√2

Oh, ok! Thank you both CPhill and Melody! 

The right angle at the bottom right sets up the following relationship

4 / y = y / 16     cross-multiply

16 * 4  =  y^2

64  = y^2     take the square root of both sides

8  = y

So....using the Pythagorean Theorem, we have that

z^2  = 4^2 + y^2

z^2  = 16  + 8^2

z^2  =  16  + 64

z^2  = 80       take the square root of both sides

z  = √80

If you want to simplify this :

z  = √ [16 * 5 ]

z = √16   * √5

z = 4√5

e^3.8  = x

Log form :

ln_e x = 3.8

b  = e

x  = 3.8

n  = x

Thank you!

#1 i believe is complex numbers

#2 is D

The triangle on the left  is a  45-45-90  right triangle....the hypotenuse  = 7√2

So....leg "a"   =   hypotenuse / √2  =  [ 7√2]  / √2  = 7

The triangle on the left is a  30-60-90  right triangle.....side "c"  is opposite the 60° angle  and  is  √3  times the side "a"

So....  c  = 7√3

a +  bi   is  ⇒    Complex

(6 + 5i)  ( 3 - 7i)  =

6 (3 - 7i)  +  5i (3 - 7i)  =

18 - 42i  + 15i   - 35i^2          (remember that i^2  = -1)....so we have

18 - 27i - 35 (-1)  =

18 - 27i + 35  =

53- 27i

4^x  = 39

Log Form :

log ₄ 39  = x

b =  4

x  = x

n = 39

a)  The data isn't perfectly linear....but it's close  !!!

If  we call  the first data point on the left  (0, 5)  and the  last data  point on the right  (45,25)

We can find the slope as    (25 - 5)  /   (45 - 0)  =  (20) / (45)  =  4/9

And  we can  use the point (20,15)   to write a  "linear"  equation

So  we have

y  = (4/9)(x -20)  + 15

y  = (4/9)x - 80/9 + 15

y  =(4/9)x + 55/9   where x  is  the number of years  since 1950

b)  The rate of change can be found by the slope  = 4/9.....this means that every 9 years.....the carbon emissions increased by about 4 billion metric tons

See my answer to the second one ( I think you already have an answer for the first )

can you explain this further with words... I don't get how you got to that conclusion

What is the y-intercept of the line that is parallel to the line y = 2x - 3 and passes through the point (6,11)?

In the form  y  = mx - b   ⇒      m  is the slope    and b  is the y intercept

So....the slope of the given line  is 2

And since parallel lines  have the same slope, the equation of our line is :

y  = 2 (x - 6)  + 11

y  = 2x  - 12   + 11

y  = 2x  - 1

So  the y intercept  is just   -1

Second one.....a (regular)  dodecagon is a 12-sided figure

We have twelve congruent triangles......if the distance between two opposite vertices of the dodecagon is 2...then  the side of one of these triangles  = 1

And the apex angle of each  of these triangles   =  360° / 12  =  30°....so...the area is just

12 (1/2) (side length of a triangle)^2 *  sin (apex angle)  = 

12 (1/2) (1^2)sin(30°)  =

12 (1/2) (1/2)  =  

12/ 4   =

3 units^2

Let d = bucket volume per unit area

Let u be rain velocity (positive downwards)

Bucket going up:    d = 2(u + V).           (1)

Bucket going down:  d = 8(u - V).        (2)

Bucket stationary:  d = t(u + 0).             (3)

Eliminate V from (1) and (2) to get u in terms of d, then substitute u into (3) and rearrange to get t.

a )

Let the cost for the frst hour  =  M

Let the cost for each  hour after that be  N

So....  the cost you paid can be modeled as

M + 1N  = 55

The cost your friend paid  is

M + 2N  = 70

Both of these are linear  because  we have equations in two variables that will result in straight lines when  plotted on a graph

If we subtract the first equation from the second we get

N  = $15......so....this must be the cost per hour after the first hour

Then  the cost for the first hour  is just     M +  15  =  55  ⇒   M  = $40

b) So....to express the total cost of a tour, C, we have

C  =  40  + 15N       whwere N is the number of hours after the first one

log 18  = x

Exponential form :

10^x   = 18

b = 10

x  = x

n  = 18

P =0.5283208.........

Q =1.4649735.........

P*Q =0.773976......., so:

8^0.773976...... = 5, so:

Log_10(5) = Log_10(8^0.773976...). Remove the "Log"

5 = 8^0.773976 =8^PQ

I get it now sorry