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8 * 10 = 80 u²

(2 * 8) / 2 = 8 u²

Using Heron's formula you can find the area of ΔABC.    [ABC] = 11.314 u²

Note that

101^2  -97^2 =  792

93^2  - 89^2  = 728

85^2  - 81^2  = 664

77^2 - 73^2  =  600

.....

13^2 - 9^2   = 88

5^2 - 1^2  =   24

Writing this in reverse, we  have  the series   24+ 88 +........ +.600 + 664 + 728 + 792

This is the  sum of an arithmetic  series

Sum =  (n/2)  [ 2a  + (n -1)  d  ] 

a = first term 

n = number of  terms which  is figured  as    (101 - 5) / 8  + 1 =      96/8  + 1  =  12 + 1   = 13

d = common difference  between  terms =  64

So....the sum  is

(13/2) [ 2(24) +  (12) (64 ) ]  =   5304

A = 343.42798714517 cm2

Perfect, as always. Thank you

240 -180  =   60°   =  pi/3

614°  =  614  - 360  =  254°  =  254 - 180  = 74°

17pi/5  =    (3 + 2/5) pi  =  3pi  +  (2/5)pi 

3pi  puts us back at  180°

So   (2/5)pi  

21pi/4  =   (5 + 1/4) pi  =   5pi  + pi/4

5pi puts us back at  180°

So  pi/4

a rectangular pyramid with length 8cm,width 5 cm, and vertical height 17 cm

tan (pi/6)   =  tan (30°)  = √3/3

sin (5pi/3 )=   sin (-60°)=  -√3/2

cos (-3pi/4)  = cos  (  360 - 135 )  = cos (225°)  =  -√2/2

So

√3/3  +  ( -√3/2) (-√2/2)   =

√3/3  +  √6/4  =      (common den  =  12  )

4√3  + 3√6

__________

      12

Jeff ordered 5 pizzas for his daughters birthday party. 60% of the pizzas were eaten at the party. When the party was over, Jeff ate 25% of what was left. If a pizza has 8 slices, how many did Jeff eat?   

Since each pizza had 8 slices                       5 • 8 = 40  slices to begin with  

At the party 60% were eaten                       0.60 • 40 = 24  slices eaten at the party  

Since 24 slices were eaten at the party        40 – 24 = 16  slices left over after the party  

Jeff ate 25% of the leftover pizza                  0.25 • 16 = 4 slices that Jeff ate after the party  

_.

(-3,-2)    (7,-2)       this will  be  a parabola that opens sideways

2a  =the distance between  the  vertices

2a = (7 - -3)  = 10

a  = 5   .....   a^2  = 25

2c = the distance between the foci  

2c = (12 - -8)

2c  = 20

c =10   .....     c^2  =100    

c^2 - a^2  =  b^2 

b^2  =  10^2 - 5^2  =  75

The center is  [ (-3 + 7)/2 , -2 ]  =  [ 4/2, -2]=  (2, -2)  =(h , k)

The equation  is

( x -h)^2         (y - k)^2

_______  -    ________  =      1

    a^2                b^2

( x - 2)^2        ( x + 2)^2

_______  -   _________  =    1

    25              75

Here's a graph :  https://www.desmos.com/calculator/jkmg428rig

(8, -2)  

vertex = (2, -1)      

Since  we have a  vertical axis of symmetry, the  form will be

y  = a ( x - h)^2  +  k

(h,k)   = the vertex  =  (2,-1)      and   we know a point....so we can find  "a"  thusly

-2 = a ( 8 - 2)^2  -1        

-2 + 1  = a (6)^2

-1  = 36a

a = -1/36

The equation is

y= (-1/36)(x - 2)^2  - 1

See the graph here  :  https://www.desmos.com/calculator/myw1qujzjc

you can keep your rude comment to yourself, and sommeone needs to learn basic manners.

116÷3√2=2n     divide both sides of the equation by two to get

n = 58 / (3 sqrt2) 

n =                   solve  using the calculator on this website

This is given by: Sn = ½ n [ 2a + (n - 1)d ]         you are given all of the inputs.....this is HOW you solve it.  (a is a₁)

what are trying to say?

Note  that   x^2  -10x - 24  =  (x - 12) (x + 2)

So  we need to  have  that  a   =  6  or  x  = -204

If a = 6    we have that  x^2  + 5x  + 6  =  (x + 2) (x + 3)   and our quotient is  (x + 3) / ( x -12)

If a  = -204  we have that  x^2 + 5x - 204  = ( x -12) (x + 17)  and our quotient is  ( x +17) / (x + 2)

So

S  = { 6 , -204 }

And their sum  =  -198

Stop trying to cheat on AoPS homework: https://web2.0calc.com/questions/hey-can-i-have-some-help-please_1

If the heptagon were inscribed in a circle, each  side  of of the heptagon  would span an  arc of  ( 360 / 7)°

The diagonals would form chords within this circle

So....using the angle of intersecting chords theorem  we have that

x  =  (1/2)  [ 2 (360/7) +  3(360/7)  ]    =  (1/2) [ 5 (360/7 ] =    [900/7]°     (  ≈  128.57°  )

x + 8 = 40

x =32

x(x + 8) =32(32 + 8) =1280 - the 2nd smallest number.

LCM(1280, 40) =1280

GCD(1280, 40) = 40

0.7......    

First....find the fractional representation of this.....

Take  the  repeating part = 7   

This is the  numerator of our fraction

For the denominator,  write the  number of  9's   equal to  the number of digits in the  repeating part =  one  9

So....  0.7.......  =   7/9

The reciprocal  = 9/7

AC = AB  = 1

CY  =   AY - AC = 6 - 1 =  5

Area of square ASHY  = (AC + CY)^2  =  (1 + 5)^2  =   6^2   = 36

Area of  2 smaller squares = 2 (1)^2 =  2

Triangles  CDY,  YEG, EFS  and SDB  are congruent,,,,each has a base of  5 and a height of 1

So....their combined area =  4 (1/2) (5) (1)   =  10

Area of DYES  = 

36 - 2  - 10   =   24

There is 22.4 l of gas in a mole of the gas (at STP).....no matter what the gas is..

   we can treat these volumes of gas as 'mole equivalents'

32.7 moles of N2 require   3 x 32.7 =98.1 moles of H2 for no reactants to be leftover....

          there is more than enough H2 to have some left over....but all of the 32.7 moles of N2 will be used

              the ratio of N2 to NH3 produced is  1 to 2

                    so there will be twice as much NH3 produced as the starting amount of N2

65.4 ml of NH3 (ammonia) produced

The diagonals of a parallelogram will bisect  each other....so we have

PT = TR       and   QT = TS

So...we have  this system

y  = 4x + 1       (1)

2y = 5x + 8      (2)

Sub (1)  into  (2)   and we have that

2 ( 4x + 1)  = 5x + 8

8x  + 2   = 5x +  8

8x - 5x  =  8  - 2

3x   =    6    →    x  =  2

And

y = 4(2)  +  1  =   9

We need to observe  the triangle inequality rule that says   that  the  sum of the lengths of any two sides of a triangle  is greater than the  length of the  remaining side

So we have

5 + 10  >  x

15 >  x

So....the  largest integral value of  x =  14

And

5 + x  >  10

x >  10-5

x > 5

So....the smallest integral value of x =  6

Sum of   intergers from  6 - 14  inclusive  =

[first integer + last ineger ]  * [ number of terms ]   / 2   =

[6 + 14]  [ 14 - 6 + 1 ]   / 2   =

[20 ] [ 9 ] / 2  =

(20/2)  * 9  =

10 ( 9)  =

90