All Answers 358090 Answers

Hint:

Use variables X and Y for the side lengths of the given rectangle.

All radii are congruent. One of the diagonal of the rectangle is a radii

So

x+4 = y + 2

x² + y² = x + 4  (Pythagorean theorem to find the diagonal)

60 + min/2 = min x 6

120 + min = 12min

min = 10.90

2 10 and 54.5 seconds

0=abs((30*2) - (5.5*M)), solve for M

M =120 / 11 =10.909090...... minutes after 2 O'Clock.

Hour hand position = 90 + min/2       (every 60 minutes the hour hand moves 30 degrees)

Minute hand = minutes x 6       (every 60 minutes the minute hand moves 360 degrees)

90 + min/2 = min x 6

180 + min = min x 12

180 + min = 12 min

180 = 11 min

16.36 = min                           3:00 + 16.36 min    = 3:16 and 21.6 sec

Ooops sorry....I went between 3 and 4 oclock!!   d'Oh !    See answers below:

The base is easy.....it's just 6 units

And  if we join  the  center of the  top circle with the  centers of the  two middle circles we will have an equilateral triangle with sides of  2r

And the  height, h, of  one of these triangles can be found as

tan 60°   =   h 

√3  = h

So....by symmetry.....the  height of this rectangle  =  2  + 2√3

So....its area = 

6  (2 + 2√3)  =

12 + 12√3  =

12 ( 1 + √3)  units^2 ≈   32.78 units^2

The triangle on the left forms  a 3 - 4 - 5 right triangle

Note that the area of  this triangle  = (1/2) (3)(4)  =   6

And drawing a segment from the lower left vertex of the large rectangle perpendicular  to the side of the shaded rectangle  will also form an altitude of this right triangle

So

(1/2) (base) (altitude)  = 6

(1/2) (5) (altitude) = 6

(5/2) ( altitude) = 6

altitude = 12/5 = 2.4

And this perpendicular  also foms a right triangle  with a hypotenuse of  3  and one leg =  2.4

So....the other leg =  sqrt (3^2 - 2.4^2 ) = 1.8

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

  (1/2) (2.4)(1.8) =    2.16 units^2

And, by AAS, this triangle is congruent to the   smaller right triangle at the  lower  right of the  figure

And we  have two of these smaller right  triangles and  two of the larger right triangles

So....their  total area =  2 ( 6 +  2.16) =  2 ( 8.16)  = 16. 32

And the area  of the large rectangle =   6 * 4 =  24

So.....the shaded area =  24 - 16.32  =  7.68 units^2

Yah....you could use that as the base (it would be 5) , but then you would have to calculate the height (which is NOT 3)...

     ___________________    ___

    /                                   /         |

  /                             3    /           h

/                                   /             |

________5_________        ___|__

Oh, I see what you mean now.  Thanks for that.  However, wouldn't the base of that parallelogram, as a hypotenuse, equal the square root of 3²+4² ?  I'm just not getting it, maybe I ought to just move on.  Thanks again for the parallelogram explanation.

Guest is referrring to the parallellgram which contains the rectangle  ....not the lareger 6 x 4 rectangle

   so area of the parallelogramI IS base x height   = 3  x 4 = 12

    I do not know about the second part of their solution though it may be correct.....

In the first place, the base of the rectangle is 6, not 3.  Second, I dispute the assumption that the inner rectangle is half the area of the larger. 

4 triangles and one rectangle.

Since z^6 - 1 = (z^2 + 1)(z^4 - z^2 + 1), the answer is n = 6 .

6 ,

The area of parallelogram is b*h=3*4=12 , Rectangle inside parallelogram and on its 2 sides (parralel sides) therefore equal to 1/2 its are so 1/2*12=6 

u^2  =  v

v^2  =  u       subtract  these

u^2 - v^2  = v  - u

( u - v) (u + v)  = v - u

(u - v) ( u + v)  =  - ( u -v)      divide both sides by (u - v)

(u + v)  =   -1   

(v + u)   =  -1

 square both sides

u^2 + 2uv + v^2  =  1

(u^2 + v^2) + 2uv   = 1

Adding the first  two equations   and we get that u^2 + v^2  = v + u....so....

(v + u)  + 2uv  =  1

-1  + 2uv  =  1      add 1 to both sides

2uv   =  2

uv  = 1

Welll...... if THAT is simplified, here is an easier way

mutiply by x/x

cube rt(4x/9x^3)    = cub rt 4x/9   / x      Multiply by 3/3

                              =    cub rt (12x/27) / x

                                 = 1/3x  cub rt 12x

                                  = \(\sqrt[3]{12x}\) / 3x                           Hmmmmmm....my answer is different than Chris'....will check

51 # / 27 q =  1.88888 # of plums per quart

1.88888 p/q x 18 q = 34 #

Equate the two functions (when the two balls are equal height)  and solve for x   (this is the TIME ...not the height)

-4.9x^2 + 12x + 2.5  =  -4.9x^2 + 14 x    

     12x + 2.5 = 14x

2x = 2.5

x = 2.5/2  = 1.25 seconds      is the TIME

   substitute this value of time into one of the equations to find the HEIGHT

       I would substitute into the OTHER equation to see if the answer is the same ....as a check of your answer !  

1 st one    Correct

2 nd one   It is the third graph down....Do you know why?  

      the parabola will be upside down due to the -3/4 leading coefficint

         then sub in some values of x to find which one fits the other  g(x)

             answer 0 and 2      I think you may have gotten confused and put in the 'y' values

                                                 ....the questin asks for the 'x' values where f(x) = g(x)

We  have  a central  rectangle with  a width of   4  and a height of 2

The area of this =  4 * 2  = 8

And we have two triangles on either end with bases of 6  and  altitudes of 2

So the areas of these are  2 (1/2) (6) (2)  = 12

So......the area of the arrow =    8 + 12     =   20

*****

See  EP's  answer  below

so 0 and 3 ? Thank you

It is the lower graph      put in x = 0   and you will see that at x= 0 value = -3

   this is also the point of intersection where f(x) = g(x)      

so you have it  1/2 correct.....    (as a side note the TOP graph also has f(x) = g(x) at x = 0   BUT they are incorrect graphs for the f(x) and g(x) given)

    BUT htere is ANOTHER Point of intersection in the lower graph where f(x) = g(x)      ....    at x = 3