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Note  that angle QRC  =  angle PRC  =  30°

So angle QRP  =  QRC + PRC  =  60°

So

QPR  =  [ 180  - QRP  -  PQR ] =  [ 180 -  60 - 65 ]  =  55°

  9 = 4 D + F

-84 =31D + F

F=205/9 First term

D=-31/9 Common difference

23rd =205/9 + (23 -1)*(-31/9)

23rd = -53 The 23rd term.                                                                                                                 

pi/4

∫       [sec x(2tanx - 5secx) ] dx  =

0

pi/4                         pi/4

∫    sec x(2tanx)      -       ∫   5 sec^2x  dx  =

0                              0

  pi/4                                       pi/4

2 ∫    sec x tanx  dx     -       5  ∫   sec^2x   dx  =

  0                                            0

             pi/4                               pi/4

2 secx ]                   -      5 tan x ]          =

            0                                    0

2 [ √2 - 1 ]    -   5 [ 1 - 0 ]  =

2√2 - 2 - 5  =

2√2  - 7

However.....this represents an area  so the correct answer  is 

abs ( 2√2  - 7 )  ≈  4.1716  units^2

Drawing an altitude from the apex to the base forms two 30-60-90  right triangles

In one of the triangles, the base  = 10   and this is the side opposite the 60° angle

The height of this triangle [ and of the larger triangle] will be opposite the 30° angle and wil be 10/√3  units long

So....the area of the biggger triangle =  (1/2)base * height =

(1/2) 20 * 10/√3  =  100/√3    sq units

Here's a pic

What's 2 to the 40th power=2^40 =1,099,511,627,776

Here's another way using just Geometry

Since   ACB is bisected......then  DC divides  AB   in a ratio of  12 /15  = BD/ DA

So.....BD  =  (12/27) * 9  =   (4/9)*9  = 4

Using the Pythagorean Theorem

CD  = √[ BD^2  + BC^2] =  √ [ 4^2  + 12^2 ]  =  √ [ 160 ]  ≈  12.65

I'm assuming that CD  is drawn to side AB

We have a Pythagorean right triangle  9 -12 - 15

Angle  ACB can be expressed  using the inverse tangent

arctangent (9/12)  =  ACB

And DCB =  .5arctangent (9/12)

Likewise, angle BDC = 90 - DCB = 90 - .5*arctangent (9/12)

So....using the Law of  Sines, we have that

CD/ sin DBC  =  BC /  sin DCB       →  DBC  = 90  →  sin 90  =  1

CD  =  12 / sin  ( 90 - .5 *arctangent (9/12) )   ≈  12.65

Here's a pic

9x-2y = -3,  - 3x+5y=27

Note that if we multiply the second equation by 3, we have

-9x + 15y = 81

So now we have

  9x  -  2y   = -3

- 9x + 15y = 81          add  these  and we have

13y  =  78             divide both sides by 13

y =  6

And to find x we can use

9x - 2(6)  = -3

9x -  12  = -3    add 12 to both sides

9x  =  9         divide both sides by 9

x =  1

So....the answer is (x,y) = (1, 6 )

Check out Khan Academy

That is 2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2 which is 2097152*x2^19

Here's how the formula is derived :

Since the radius, r, of the incircle can be drawn to meet each side of the larger triangle at right angles, then we can construct three triangles by connecting each vertex of the larger triangle to its incenter.  And each of these triangles has a height of r and a side as a base.

So.....the total area, A, of the the larger triangle is

(1/2) r ( a + b + c)  =  A      where a,b,c are the side lengths

Multiply both sides by 2

r (a + b + c)  = 2A

Divide both sides by (a + b + c)

r =  2A / (a + b + c)

But (a + b + c)  is the perimeter = P

So....

r =  2A / P

281.25 / 45  = 6.25  =   625%

Note that

2.4^2  =  5.76 → √ 5.76 =  2.4

2.5^2  = 6.25   → √6.25  = 2.5

2.6^2  = 6.76   → √6.76  = 2.6

2.7^2  = 7.29  → √7.29  = 2.7

So

√ 5.76 + √6.76  =  2.4 + 2.6  = 5.0    (1)

And

√6.25 + √7.29  =  2.5 + 2.7  =  5.2    (2)

But

(1)  < √6 + √7 <  (2)  

So  √6 + √7  must lie between  5.0  and 5.2

Thus.....√6 + √7  must lie between  5 and 6

OK....post your questions

62.5  

Add a 0, move the decimal two places to the right and write a % sign  =

6250%

The radius  of the largest circle that will fit inside the triangle is given by :

2A / P

Where A is the area of the triangle and P is the perimeter

To calculate the area we can use Heron's formula

s =  the semi-perimeter  =   [ 5 + 6+ 7] / 2  =  18/2  = 9

And the area is given by

√ [ s (s - a) (s - b) (s - c) ]     where a, b, c are the side lengths of the triangle

So we have

√ [ 9 (9 - 5) (9 - 6) (9 - 7) ]   = √ [ 9 * 4 * 3 * 2 ]  = √[36 * 6 ]  = 6√6

And the perimeter  is 18

So.....the radius is  given by

2 [ 6√6] / 18  = 

(2/3)√6

pretty sure its 90.6 bud

We can use the concept of "Egyptian" fractions, here

25/27

Take the ceiling function for the division of 27 by 25  =  2

Write this as the denominator of a fraction with a numerator of 1 =  1/2

Subtract this from 25/27

25/27 - 1/2  = 23/54

Take the ceiling function for the division of 54 by 23  =  3

Write this as the denominator of a fraction with a numerator of 1 =  1/3

Subtract this from 23/54 - 1/3  = 5/54

Take the ceiling function for the division of 54 by 5  =  11

Write this as the denominator of a fraction with a numerator of 1 =  1/11

Subtract this from 5/54

5/54 - 1/11  = 1/594

We can now stop the process since the last subraction results in a unit fraction

So.....25/27  can be represented  as the sum of these unit fractions

1/2  +  1/3 + 1/11 +  1/594

BTW.....this is the shortest length possible...in other words.....this fraction cannot be represented by  the sum of any fewer than 4 unit fractions

15/16

Using a similar process this can be represented as the sum of the following unit fractions :

1/2 + 1/3 + 1/10 + 1/240

Again....this fraction cannot be represented by the sum of any fewer than four  unit fractions

Here is website devoted to Egyptian Fractions....it is rather interesting :

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html

We are just substituting the values into the function for x and evaluating

For example

f(0)  =  3(0)^2  - 4(0)  + 5  =   0 -  0 + 5  =  5

See if you can do the other two

The time is given by

(75/170)(3)   ≈ 1.32 hours  ≈ 1 hr 19 minutes

I've found my error at -3-5=7, the result should be 8/9 as the site shows.

It means there are 2 answers.

For one answer you add and for the other one you subtract :)

Yes, Vietnamese :)

Thanks man, the result i got was x>3/-5, looks like i misplaced the minus. (bad english).

2*(x+3) >3*(1-x)    simplify

2x + 6 > 3 - 3x      add 3x to both sides, subtract 6 from both sides

5x >  -3                divide both sides by 5

x >  -3/5