CPhill

Nice, tertre  !!!

Parallel lines  have the same slope

So...we have

y  = m  (x - x1)  + y1      where m = the slope  = -3/4   and  (x1, y1)  = ( 4,2)

y  = (-3/4) ( x - 4)  + 2

y  = (-3/4)x + 3 + 2

y = (-3/4)x + 5      [in slope-intercept  form ]

Perpendicular lines have negative reciprocal slopes.....so....the slope of the line we want will be -7/2

So  we have

y  = m ( x - x1)  + y1          where  m is the slope and  (x1, y1)  = ( 2,4)

y  = (-7/2) ( x - 2)  + 4         simplify

y = (-7/2)x + 7 + 4

y = (-7/2)x + 11    [  in slope-intercept form ]

We have an irregular trapezoid

If we drew a perpendicular line from the top right vertex of the figure to the base......the right half of   the two figures formed would be a right triangle 

The base length of this triangle  is (24 - 10)  = 14  units

And since one of the angles of the right triangle is 45.....the other angle at the top of the triangle is also 45....and in a 45-45-90  right triangle.....the sides opposite the 45° angles are equal....so....the height of this triangle  = the base length  = 14  =  the height of the trapezoid

And the area of a trapezoid  is 

(1/2) height  *  ( sum of the bases)  =

(1/2) (14) (24 + 10)  =

7 * 34  =

238

Here is a picture to help you understand

We have an octagon  with an  apothem  of 8

We have a right triangle formed

The central angle formed by the triangle is 22.5°

So...the base of this triangle  (1/2 side length of the octagon)  is given by 8 tan (22.5°)

The apothem and the base therefore form two legs of the right triangle

And the area of a right triangle  = (1/2) product of the leg lengths

So....the area  of the right triangle is  (1/2) (8) (8 tan 22.5°)

But  we  have  16 of these  congruent right triangles that comprise the area of the octagon

So...the total  area is   (16) (1/2) (8) * 8 tan (22.5°)  =

8^3 * tan (22.5°)  ≈  212.1  units^2

y  = 0.32x + 20.51

If x  = 175, we have

y  = 0.32 (175) + 20.51  

y  = 76.51   ≈  77 hits

(25x^3 - 10x^2 + x )

_______________         ≥ 0

      8 - x^3

x ( 25x^2 - 10x  + 1)

________________       ≥  0

(2 - x) ( x^2 + 2x + 4)

x ( 5x -1) ( 5x - 1)

_________________      ≥  0

  (2 - x) (x^2 + 2x + 4)

x (5x - 1)^2

________________        ≥  0

(2 - x)(x^2 + 2x + 4)

Note  that   (5x - 1)^2   is  ≥ 0  for all x   and  ( x^2 + 2x + 4)  is > 0  for all x

So   we only need consider what  x values makes

  x 

_____     ≥ 0

2 - x

Note that  if    x  is negative...the function is negative

And if  x > 2, the function is negative

So...the x values that make this ≥  0    are   0 ≤ x < 2 

I'm not that good at these, but I will give this one a try

Note that, for the given constraints, as small as the first function can be for any x  is  -4

So.....any value we choose for x  will be evaluated as being ≥ -4

So...in the first case....we are looking for  this

(x^2 - 4)^2  - 4  = 5

x^4 - 8x^2 + 16  - 4  = 5

x^4  - 8x^2 + 7  = 0

Factoring, we have

(x^2 - 1 ) ( x^2 - 7)  = 0

Setting both factors to 0 and solving for x  results in the values    1, -1 ,  √7 , - √7 

All of these make f (f(x))  = 5

In the other case, we need to choose an x < -4  which we can put into  x + 3  which will result in 3  or -3....note that when we put  -3 or 3 into the first function we get  5

0 will make x + 3  = 3  but x has to be < -4 to use this function

But  -  6  will work  because   f(-6)  =  (-6) + 3  =  -3

And this value when plugged into the first function to give a result of 5

That is  f ( f(-6) )  =  5

So...the values for x  that  will work are   1, -1 ,  √7 , - √7   and  - 6

At a 9% off sale Alexandra bought a new drill and saved $17. How much did she pay for the drill? (two decimal places)

Call the original price, P

9%  off  = 91%  of the original price  = .91P

And a savings of $17  = P - 17

So we have 

P ( .91)  = P -  17     add 17 to both sides,  subtract .91P from both sides

17  =  1P  - .91P

17  = .09P     divide both sides by .09

$188.88  = P  (the original price )

Richard's furniture store bought a couch and marked the price up 31%. At a 11% off sale the couch sold for $2,820. How much did Richard pay for the couch originally? (two decimal places)

Call the original price that Richard paid, P

The price after the markup was  (1.31) P

After an 11%  markdown the price was  ( 1.31 P ) (.89)

So we have

(1.31 P ) (.89)  = 2820

1.1659 P = 2820       divide both sides by  1.1659

P  = 2820 / 1.1659  =  $2418.73

The  population after a number of years  is given by :

Beginning population* (1 +  percentage growth rate expressed as a decimal)^(no. of years) =

48314  ( 1 + .035)^32  =   145266  people after 32 years