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x - y = sqrt (3)

x + y = 2              add these

2x  = 2 + sqrt (3)

x =   [ 2 + sqrt (3) ]  /2

y =     2 - x

y =  2 - [2 + sqrt (3) ] / 2  = ( 4 - [2 + sqrt 3] )  / 2  =     [ 2 - sqrt (3)] / 2

xy  =   [ 2 + sqrt (3) ]  /  2   *    [2 -sqrt (3) ] / 2  =     (4 - 3) / 4 =   1/4

The common  ratio  =  5/a

The  sum  =          4 / ( 1 -5/a)  =  (4)  /  [ ( a - 5) / a ] =     (4a) / (a - 5)

The smallest integer value of  a  that produces a perfect square is when   a =   9

Angle B  =  180 - 53 - 44 =   83°

By the Law of Sines

BC  /sin A = AC /sin B

BC = AC sin A / sin B

BC  =   11* sin (53°) / sin (83°)   ≈    8.85

Do not post A o P S homework.

it is wrong

(-8, 8)  should be on  both lines.......but it cannot be on the  first line  because

-8 + 5(8)  =  32

But  this should =  7

7! + 6! =   5760

1 | 2 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 12 | 15 | 16 | 18 | 20 | 24 | 30 | 32 | 36 | 40 | 45 | 48 | 60 | 64 | 72 | 80 | 90 | 96 | 120 | 128 | 144 | 160 | 180 | 192 | 240 | 288 | 320 | 360 | 384 | 480 | 576 | 640 | 720 | 960 | 1152 | 1440 | 1920 | 2880 | 5760 (48 divisors)

Count 'em up  !!!!   [easiest to  just count the odd divisors and  subtract this number  from  48   ]

The slant height of each cone =  the radius of the original circle 6 in

The circumference of  each  cone at the base =  Circumference of the  circle /  4 =   (2pi) (6)  / 4 =     3 pi

C = 2pi r

3 pi  =  2pi * r                  divide both sides by 2 pi

3/2  = r =   1.5 in

The height of each cone  = 

sqrt  [ slant height ^2  - radius^2 ] =

sqrt  [ ( 6)^2   -  1.5^2 ] =    sqrt   [ 36 - 2.25 ]  ≈  5.8  in 

See the graph here : https://www.desmos.com/calculator/k4kxivfue1

This forms a  triangle  with a base =  8 - - 6 =  14

And a height of 2 + 1/3  =  7/3

The area is   (1/2) (14) (7/3)   = 49 / 3  units^2

The slope is

[ 2 - 7 ] / [ 3 - -1 ] =  -5 /4

Using the second point and the  slope to  write the equation of the line

y= -(5/4) (x -3) + 2

When x = -6

y = (-5/4) ( -6 - 3) +  2

y = (-5/4)(-9) + 2

y = 45/4  + 8/4

y =  53 / 4

See alan's answer here :  https://web2.0calc.com/questions/please-help_63707

12, 15, 18, 21, 51 , 81, 102 , 105 , 108, 111

The sum of the roots =  -(-11) / 5 =  11 / 5

So

1 / (a + b)  =  1 / (11/5)  =   5 /11

t^2 -9t - 36 + t^2 - t + 24         simplify

2t^2 - 10t - 12

The t that produces the minimum is     -(-10) / (2 * 2)  = 10 / 4  =  5 / 2  =  2.5

sumfor(n, 2, 6, 1 / (n^3 - n)==It converges to 5 / 21

Sorry Melody! I won't do it again. 

Recall vertex form: \(a(x-h)^2 + k = y\)

Substituting the vertex, we have: \(a(x + 4)^2 = y\)

Plugging in the point, we have: \(25a = -7\).

Can you take it from here?

Here   :   https://web2.0calc.com/questions/geometry_57153

Circumradius =   product of the sides  / [ 4 * area  ]

Product of the  sides = (29)^2 * 40  =  33640

Height of triangle =  sqrt (29^2 - 20^2)  =   sqrt (441)  = 21

Area =  (1/2) base * height =   (1/2) (40) (21)  = 420

4*area =  1680

Circumradius =  33640  /  1680   ≈   20.02

The parabola tuurs upward and  has roots at   x = -2  and  x = 2

So  we have

(x + 2) (x - 2) > 0

x^2 - 4  > 0

b =  0

c =  -4

b + c =  0 - 4  =  -4

Set the equations equal

2x^2  - 7 =  -x^2  - x + 1     simplify

3x^2 + x - 8   =  0

Using the quadratic formula

x   =     -1   ±√ [1^2 - 4 (3) (-8) ]                -1 ±√ 97

            ____________________ =        ________

                         2 * 3                                     6

c - a  =  2√97 / 6  =    √97  /   3

                                    Cracked           Not Cracked          Total

One Peanut                    18                      54                       72

Two Peanuts                   26                     71                       97

Three Peanuts                                           5                         5

                                     ________________________________

                                        44                    130                     174   

P(shell is  cracked )=  44 / 174 =    22 / 87 

P(not cracked and  contains 2 peanuts)  = 71 / 174

P (not cracked or  contains 2 peanuts  ) =  P(not cracked) + P (2 peanuts) - P(not cracked and 2 peanuts)  =

(130) / 174   +  (97)/174  = (71) / 174 =   156 / 174  =  78 / 87  = 26 / 29 

P not cracked given 2 peanuts)  =   71 / 97

Please do not 'make people's lives easier' by answering over me Builderboi.

If guest had have got back with some suggestion of doing some work for themselves then by all means, help some more.

But I asked thei guest to come back and show that they had done something for themselves and I do not want other answerers answering over me.

Note:

If I ever answer over you and you  take exception to what I have done then by all means tell me that you are not happy.

I do not expect this to be a one way street.

Anyway, yoiu cannot just cancel out x+3 because sometimes it is negative and somteimes it is positive.

This is what I would do...

\(\frac{dy}{dx}=\frac{2x}{3y}\\ \frac{d^2y}{dx^2}=\frac{(3y)(2)-(3\frac{dy}{dx})(2x)}{(3y)^2}\\ \frac{d^2y}{dx^2}=\frac{(y)(2)-(\frac{dy}{dx})(2x)}{3(y)^2}\\ \frac{d^2y}{dx^2}=\frac{2y-(\frac{2x}{3y})(2x)}{3y^2}\\ \frac{d^2y}{dx^2}=\frac{2y}{3y^2}-\frac{4x^2}{9y^3}\\ \frac{d^2y}{dx^2}=\frac{6y^2-4x^2}{9y^3}\\ \)

OR

\(2x^2 - 3y^2 = 4\\ 4x-6y\frac{dy}{dx}=0\\ 4-6[y(\frac{d^2y}{dx^2})+(\frac{dy}{dx}\cdot \frac{dy}{dx})=0\\ 4-6[y(\frac{d^2y}{dx^2})+\frac{2x}{3y}\cdot \frac{2x}{3y})=0\\ 4-6[y(\frac{d^2y}{dx^2})+\frac{4x^2}{9y^2}]=0\\ 4-6y(\frac{d^2y}{dx^2})-\frac{6*4x^2}{9y^2}=0\\ 3y(\frac{d^2y}{dx^2})=2-\frac{4x^2}{3y^2}\\ 3y(\frac{d^2y}{dx^2})=\frac{6y^2-4x^2}{3y^2}\\ \frac{d^2y}{dx^2}=\frac{6y^2-4x^2}{9y^3}\\\)

LaTex

\frac{dy}{dx}=\frac{2x}{3y}\\
\frac{d^2y}{dx^2}=\frac{(3y)(2)-(3\frac{dy}{dx})(2x)}{(3y)^2}\\
\frac{d^2y}{dx^2}=\frac{(y)(2)-(\frac{dy}{dx})(2x)}{3(y)^2}\\
\frac{d^2y}{dx^2}=\frac{2y-(\frac{2x}{3y})(2x)}{3y^2}\\
\frac{d^2y}{dx^2}=\frac{2y}{3y^2}-\frac{4x^2}{9y^3}\\
\frac{d^2y}{dx^2}=\frac{6y^2-4x^2}{9y^3}\\

Here, I'll make your life a little bit easier:

We have: \((2x+10)(x+3)<(3x+9)(x-8)\)

Now, factor \(3x + 9\): \((2x + 10)( x+3) < 3(x + 3)(x-8)\)

Note that because both sides have an \(x + 3\), we can cancel them, giving us \(2x + 10 < 3(x-8)\).

To finish, solve it like you would a regular equation, but make sure the sign( < or >) satisfies the equation.