CPhill

That's a nice way to see this, guest.....!!!!

OK......I hope this is a big enough  "PI" for eveyone to share....as you might have noticed.....I've already had a slice.....!!!!

The right angle is PQR

Note that we can consider   PQ to be the  base   and QR to be the height

PQ  =  √ [  6^2  + ( 9 - 5)^2 ] = √ [6^2 + 4^2 ] = √ 52  = 2√13

QR  =√ [ (12 - 6)^2 + 9^2 ]  = √ [36 + 81 ]  = √117 =  3√13

So....the area  =   (1/2) base * height    =   (1/2) * (2 √13) * (3√13)  =  3*13  units^2  = 39 units^2

Thanks to AZSun for catching my error  !!!

We are looking for an integer whose sum of its digits  = 1274

Here's a possibility :

S(n)  = 1274  = S ( 7777....7)   where we have 7 repeated 182 times

Note that    7 * 182  =   1274

So....  S ( n + 1) = S( [7777....7] + 1 )  = S(1275)    =  S (555.....5)   where we have 5 repeated 255 times

Note that  255 * 5  =  1275

Well, X².....by 2042, I'll probably be dead....so....you may catch me yet  !!!

f(x)  = 4x^(1/2)   - x

f ' (x)   =  (1/2*4x^(-1/2)  - 1  =    2x^(-1/2)  - 1

The tangent line will be horizontal if  the slope  = 0

So.....set the derivative equal to 0 and solve for x

2x^(-1/2)  -  1  = 0

2x^(-1/2)  = 1 

x^(-1/2)  = 1/2

We can write this as

x^(1/2)  = 2        square both sides 

And  x  =  4

When  x  =  4 ,   y  = 4√4  - 4   =  8 - 4  =  4

So.....the point where the tangent line is horizontal is  (4,4)

To find where the slope is  -1/2....we have

2x^(-1/2)  -  1  = -1/2

2x^(-1/2)  =  1/2

x^(-1/2)  =  1/4

And we can write this as

x^(1/2)  =  4     square both sides

x  = 16 

When  x  = 16 ,   y  = 4√16  - 16   =  0

So....the point where the tangent line  = -1/2  = (16, 0 )

If we have a multiplicity of 2 at  (-1,0)  and the point (7,0) is also on the graph, we must have that

P(x) =

a (x + 1)^2 (x - 7)      

a (x^2 + 2x + 1) (x - 7)  

a(x^3 + 2x^2 + x -7x^2 -14x - 7)  

a(x^3 - 5x^2 - 13x - 7)  

And if   (0, -14)  is on the graph.....we have that

-14  = a ( 0^3 - 5(0)^2 - 13(0)  - 7)

-14 = -7a

a  = 2

Here's a graph : https://www.desmos.com/calculator/wbejrdzi2x

2.  We  can find  P  with the tangent inverse  (other inverse functions are also possible )

arctan (12/5)  = P = 67.38° 

3.  C  = arctan (22/120) = 10.39°

4.  We know the adjacent side to L  and the hypotenuse.......the cosine inverse is the way to go, here

arccos ( 18/60)  = L  = 72.54°

5.   We know an angle an opposite side and we are looking for the hypotenuse

So.....the sine function is what we need

sin (62)  =  45 / hypotenuse     rearrange as

hypotenuse  = board length  =  45 / sin (62)   =  51 in

Last one :

Area of sector   GHJ  =

(1/2)  radius^2 *  measure of angle GHJ in radians.....so we have...

(1/2)  (5^2)   * pi /4  =

25 / 8  pi  cm^2   ≈   9.8 cm^2    

Second one....

Here's a YouTube video that explains this better than I can.....

https://www.youtube.com/watch?v=YEX2mhvbYuo