CPhill

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 )

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

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

We can use the same idea here, tertre, as we did the other day on this type of problem

The line of symmetry divides the shaded area into equal halves

If we only consider the left half.....we will have the sector of a circle QRS  which will be (60° /360°) =  1/6 the area of a circle with a radius of 12  = (1/6) pi (12)^2   =  144/6  pi     =  24pi   units^2   (1)

Now....cosider triangle QRS...its area is  (1/2) (12)^2  sin (60°)  = 72√3 / 2   =   36√3  units^2   (2)

So   1/2   of the shaded area  = (1)  - (2)  =

[24 pi - 36√3]  unts^2  =    12  [ 2pi - 3√3 ] units^2

So....twice this is the shaded region's area  =   24 [ 2pi - 3√3 ] units^2