CPhill

1225 = 1000 (1 + r)^3     divide both sides by 1000

1.225  = (1 + r)^3          take the cube root of each side

∛ 1.225   =  1  + r        subtract 1 from both sides

∛1.225  - 1   =  r   ≈    0.07  =  7 %

Johann has to work on the following task during his apprenticeship as a stonemason: "From a rectangular sandstone slab measuring 150 cm x 100cm x 6cm (length x width x depth), a corner broke off due to a transport accident.The break point has already been straightened out a bit this plate the largest possible rectangular table top.

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Since the width is 100cm, it appears that 50 cm of the length broke off during transport......So.....The largest rectangular top that can be made  would be 100cm X 100cm  = 10,000 cm^2

{ The depth of the top doesn't matter  ]

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Da die Breite 100 cm ist, scheint es, dass 50 cm der Länge während des Transports abbrachen ...... Also ..... Die größte rechteckige Oberseite, die hergestellt werden kann, wäre 100 cm × 100 cm = 10 000 cm ^ 2

{Die Tiefe der Spitze spielt keine Rolle]
 

The slope of this line is    [ 3 - 0 ] / [ 0  -  -8 ]  =  3 / 8

So....using  (0,3,), the equation of the line is

y  = (3/8) x  + 3

And using (t, 5),  t  can be found thusly :

5 =(3/8) t  + 3     subtract 3 form both sides

2  = (3/8)t      multiply both sides by 8/3

16/3  =  t

Put A at (0,2), C at (0,0), and B at (6,0)

AB  is   √40  units in length

Construct two lines perpendicular to AB  through A  and B

The equation of the line through A  is  y = 3x + 2 .......the of the line through B is

y = 3x  - 18

Constuct two circles with radiuses of √40  centered at A  and B

The equation of the circle centered at A  is  x^2 + (y - 2)^2  = 40

The equation of the circle centered at B  is  (x - 6)^2 + y^2  = 40

And we can find the intersection of the circle centered at A and the perpendicular line to AB passing through A  thusly :

x^2 + (3x + 2 - 2)^2   =  40

x^2 + 9x^2  =  40

10x^2  =  40

x =  2   and y  = 3(2) + 2  = 8

So  the intersection is at  (2, 8)....label this point D

Likewise....we can find the intersection of the cirlce centered at B and the perpendicular line to AB passing through B  thusly :

(x - 6)^2  + (3x - 18)^2  = 40

(x - 6)^2 + 9 ( x  - 6)^2  = 40

10 (x - 6)^2  = 40

(x - 6)^2  =  4

x - 6  = 2

x = 8    and y  = 3(8) - 18   =  6

So the intersection  is  (8, 6).....label this point E

And  DB  will be one diagonal of the square and AE will be the other diagonal

And the equation of DB  is   y  = -2x + 12

And the eqation of AE  is y =  (1/2)x  + 2

And we can find the inersection of the diagonals, thusly

-2x + 12  = (1/2)x + 2

10 = 2.5x

4  = x      and  y  =  -2(4) + 12   = 4

So the intersection of the diagonals is at (4,4)

And the distance between this point and C  =  √ [ 4^2 + 4^2]   = √32  = 4√2

The side of the square  =    diagonal length / √2

So....the side length  =  [10√2] / √2   =  10cm

So....the area = side^2   =   10^2   =  100 cm^2

The answer is "B"....I'll fill in the blanks, later.....

If x^2 - 4  is a factor.....then  (x - 2)  and (x + 2)  are  factors

Which means that   2  and - 2 are roots

So

4(2)^4+7(2)^3 +a(2)^2 +b(2) +8  = 0

4(-2)^4+7(-2)^3 +a(-2)^2 + b(-2) +8  = 0

Add these and we have that

128   +  8a   +   16  = 0    ⇒  8a  =  -144  ⇒   a  =  - 18

Subtract them and we have that

112 + 4b   =   0

28 + b  =  0    ⇒   b  = -28 

So....the polynomial is

4x^2  + 7x^3  - 18x^2 - 28x  + 8       and we can write this as

[4x^2  + 7x^3  - 2x^2]  -  16x^2  - 28x + 8  

x^2 (4x^2  + 7x - 2)  -  4 ( 4x^2 + 7x - 2 )

(x^2 - 4) (4x^2 + 7x - 2)

(x^2 - 4) (4x - 1) (x + 2) 

(x - 2) (x + 2) (4x - 1) (x + 2)  =  0

The  values that make this  = 0   are     x= 2, x = 1/4  and x  = -2

If x - 2 is a factor....then  2  is a zero

Thus  

2(2)^3  - a(2)^2  + 5b(2) + 4b  = 0

16 - 4a + 10b  + 4b  = 0

-4a + 14b  = -16

2a - 7b  = 8                   (1) 

And

2(-1)^3  - a(-1)^2  + 5b(-1) + 4b  = -15

-2   - a   - 5b + 4b   = -15

-a - b  =   -13   ⇒   -2a - 2b  =  -26      (2)

Add (1)  and (2)    and we have

-9b  =  -18    ⇒    b  = 2

So

-a - 2  = -13

-a  =  -11

a  = 11

So....the polynomial is

2x^3 - 11x^2 + 5(2)x + 4(2)

2x^3 - 11x^2 + 10x + 8

And we can write

2x^3  [ - 4x^2  -  7x^2 ] + 10x + 8

[2x^3 - 4x^2]  - [7x^2 - 10x - 8 ]

2x^2 (x  - 2)  -  [ 7x^2 - 10x - 8 ]

2x^2 (x - 2) - [ ( 7x  + 4) (x - 2) ]

(x - 2) (2x^2 - 7x - 4)

(x - 2) (2x + 1) ( x - 4)

Sorry.....I believe that exactly 2 should be   6 + 6 +  12   =   24 

a + 980   = 2000    subtract  980  from both sides

a  =  2000 - 980   =  1020