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Tan 45 = sin45/cos 45        cos and sin are = at 45 degrees    so tan 45 = 1

3^11  + 5^7  +  1  =  255273

Sum of the  digits  =  24    which is divisible  by  3

So......   3   divides  255273  =   the  smallest prime divisor

3x^2  +  4x  +  12  =  x^2   - 5x  +  7         rearrange  as

2x^2  + 9x   +  5    =  0

The  sum  of the  roots =  -9/2  =  r + s                square  both sides

r^2  +  2rs  +  s^2   =   81/4

The product of  the roots =  5/2   =  rs

So  ......  2rs   =  5

So

r^2   +  2rs   +  s^2   =    81/4

r^2  +   5  +   s^2    = 81/4

r^2   +  s^2    =  81/4 -  5    =  81/4  -  20/4   =   61 / 4

Hey there, Guest!

So, let's go through this problem step-by-step:

First, let's format it like this: \(\frac{x+1/x}{x-2/x}=\frac{5}{4}\)

\(\frac{x^2+1}{x^2-2}=\frac{5}{4}\)

Step 1: Multiply both sides by \(x^2-2\).

\(x^2+1=\frac{5}{4}x^2+\frac{-5}{2}\)

\(x^2+1-\frac{5}{4}x^2=\frac{5}{4}x^2+\frac{-5}{2}-\frac{5}{4}x^2\) (Subtract \(5/4x^2\) from both sides.)

\(-\frac{1}{4}x^2+1=\frac{-5}{2}\)

\(-\frac{1}{4}x^2+1-1=-\frac{5}{2}-1\) (Subtract from both sides.)

\(\frac{-\frac{1}{4}x^2}{-1/4}=\frac{\frac{-7}{2}}{-1/4}\)

\(x^2=14\)

Therefore \(x=\sqrt{14}\) and \(-\sqrt{14}\).

Hope this helped! :)

( ﾟдﾟ)つ Bye

Thank you so much 😊

A

10                  M

B               20                   C

AC =  sqrt  (10^2  +  20^2)  = sqrt  ( 500)    =  10sqrt 5.....so........AM =   5sqrt 5

BM  =       (10)(20)  / (5 sqrt  5)     =   40/sqrt 5  =   8 sqrt  5

Using  the Law of Cosines

AM^2  =  AB^2    +  BM^2   -2 (AB * BM)  cos  ( ABM) 

125  =  100  +  320    - 2   ( 10 * 8sqrt 5 )  cos (ABM)

(125  - 100   -320 )   /  ( -160sqrt 5)   = cos (ABM)

( 59 sqrt 5)  / 160    = cos (ABM)

9  times

10, 20,30......90

 (x + 1/x):(x - 2/x) = 5:4

x + 1/x  =    (x^2 +1)  / x               (1)

x - 2/x    =   (x^2 - 2)  / x               (2)

Diviiding (1) by (2)   we  have

x^2 + 1            5

______    =    ___                  cross-multiply

x^2  - 2            4

4x^2  +  4     =    5x^2  -  10        rearrange  as

x^2   -   14   =   0

x^2  =  14                take both roots

x = sqrt 14                   x =  -sqrt  14

Edit:

I'm wrong, the raduis is 12, not 6. 

=^._.^=

As follows:

For a circle?
 

(x-0)^2    +    ( y-5)^2   = -13 + 25

(x-0)^2     + (y-5)^2    = 12                             circle with center   o,5    radius   sqrt 12

See image below

1/4 of the circle    minus the triangle is 1/2 of the shaded area

1/4 of the circle    1/4 pi (12^2)  = 36pi

area of triangle   1/2 *  12 * 12 = 72

36 pi  - 72      * 2     =   72pi - 144 = ____________ units²

x + 5y = 7 does not contain the point (8, 3). 

=^._.^=

I really need to figure out how to make diagrams.... sorry if this is confusing. 

Notice that 1/2 of the grey space is 1/4 of a circle - the right triangle. 

So the total area of the grey space is 2(6^2*pi/4 - 6*6/2).

2(6^2*pi/4 - 6*6/2) = 18pi - 18

=^._.^=

For a certain value of \(k\), the system
\(\begin{align*} x + y + 3z &= 10, \\ -4x + 3y + 5z &= 7, \\ kx + z &= 3 \end{align*}\)
has no solutions. What is this value of \(k\)?

\(\begin{array}{|rcll|} \hline \begin{vmatrix} 1 & 1 & 3 \\ -4 & 3 & 5 \\ k & 0 & 1 \\ \end{vmatrix} &=& 0 \\\\ k* \begin{vmatrix} 1 & 3 \\ 3 & 5 \\ \end{vmatrix} +1* \begin{vmatrix} 1 & 1 \\ -4 & 3 \\ \end{vmatrix} &=&0 \\\\ k*(1*5-3*3) + 1*(1*3-(-4)*1) &=& 0 \\\\ k*(-4) + 1*7 &=& 0 \\\\ k*(-4) &=& -7 \\\\ k &=& \dfrac{-7}{-4} \\\\ \mathbf{k} &=& \mathbf{ \dfrac{7}{4} } \\ \hline \end{array}\)

-x^2 +8x-4       is a dome shaped parabola    the max will occur (at the vertex) at the value of   x = - b/2a    where b = 8   a = -1

                               use this value of 'x' in the equation to calculate the maximum value of the equation

no real solutions means the discriminant is < 0

b^2 - 4ac <0

k^2 - 4(5)(88) <0

k^2 < 1760

k <  41.95      k > -41.95            Largest integer would be 41

x - y = a

x = a + y

2(x^2 + y^2) = x + y + 1

2((a + y)^2 + y^2) = (a + y) + y + 1

2(a^2 + 2y^2 + 2ay) = a + 2y + 1

4y^2 + 4ay + 2a^2 = a + 2y + 1

(4)y^2 + (4a-2)y + 2a^2 - a - 1 = 0

We're looking for the highest possible a value. 

This equation is only true when the discriminant is non negative. 

b^2 - 4ac >= 0

(4a-2)^2 - 4(4)(2a^2 - a - 1) >= 0

-16a^2 + 20 >=0

-4a^2 + 5 >= 0 

-4a^2 >= -5

a^2 =< 1.25

So the max value of a is sqrt(1.25).

This could probably be done faster geometrically too. 

=^._.^=

Volume of a sphere   = 4/3 pi r^3

there are 50 of them   soo      50 * 4/3 pi (4^3) = 13404.12866 m^2

volume of cube = l * w * h      (the box in the question is not a cube....l   w   and h are not the same value)

                                               2 * 6 * x = 13404.12866

                                                     h = 1117 m                              ( are you sure you have the dimension of the spheres correct?)

Don't worry, all is well between us.

The acceleration is just gravity. (for falling objects)

It is the rate at which something is increasing or decreasing in velocity.

Maybe you will work it out better with time.  It is a difficult concept.

The inside angles of a triangle sum to 180 degrees    29 + 82 + 'other angle' = 180

         'other angle' = 180 - 29 - 82        solve for 'other angle'

then:

'other angle'  + x    form a straight line

'other angle' + x = 180     solve for 'x'

The three angles sum to a straight line    180 degrees

42 + x + 29 = 180 degrees    solve for x

The center (h,0 ) is the same distance from each of the points    (h,0) because it is on the x axis then y = 0

  distance from point 1 to center h,0

      sqrt [ (h-2)^2 + ( 0-3)^2 ]

  distance from point 2 to center

      sqrt [ (h-3)^2 + ( 0-2)^2 ]                  Now you have two equations, two unknowns ...solve for h,k

sqrt [ (h-2)^2 + ( 0-3)^2 ]   =     sqrt [ (h-3)^2 + ( 0-2)^2 ] 

(h-2)^2 + (0-3)^2     =     (h-3)^2 + ( 0-2)^2    

h^2 -4h + 4      + 9      =   h^2 -6h+9     +  4 

-4h + 4           + 9              =  -6h + 9       + 4

2h     +13                      =     13

2h  = 0

h=0

So the center is 0,0        can you calculate the radius now?     Need more help?     (use distance formula from 0,0 to either of the points)

Thanks Heureka,

That is a really neat solution