CPhill

Let x be the cost of a shirt  and y be the  cost of a tie.....so....we have this system of equations :

6x + 3y =  79.50

3x  + 2y = 41

Multiply the second equation through by -2  and we have

6x + 3y =  79.50

-6x - 4y  = -82        add these 

-y=   -2.5

y= 2.5   = $2.50 = the cost of a tie

And using   3x + 2y  =41  we can find the cost of a shirt

3x + 2(2.50) = 41

3x  + 5 =  41         subtract  5 from both sides

3x = 36   divide both sides  by 3

x = $12  = the cost of a shirt

Wow!!!!....that's impressive !!!!!!

I'm going to have to study this one   !!!!

cos  ( arcsin (-0.845))

Easy way : 

arcsin (-0.845) = - 57.67°

cos (-57.67°)  ≈ .535 

"Expert" way

We are looking for  the cosine  of an angle whose sine =  -845/ 1000 =  y  /r

Remember that cosine  =  x /r

We need  to  find x  

x =  ±√ [ r^2 - y^2]  = ±√ [ 1000^2 - 845^2 ] = ±534.76 = ±535

This angle is in the 4th quadrant  (the sine is negative there).....but  the cosine is positive in this quadrant

So

cosine  for this angle =  x / r  =  535 / 1000   =  .535

The wires form the hypotenuses of two right triangles.....we need the distance in each case of the opposite side to the 75° angles

sin (75)  =   distance₁ / 12

sin (75)  =   distance ₂ / 16

So....We can find the distance thusly :

(12 + 16) sin (75)  =   distance₁ + distance₂

28sin(75) ≈ 27.05 ft

First one

Is not one-to-one

(-pi/2, pi/2)

Second one....Correct  !!!

(b) What is the probability that we draw any straight (including "straight flush" and "royal straight flush" hands)?

This is similar  to the last one

If  we let  the ace be the lowest or highest card in the straight.....we have 10 possible straights within the ranks :

A-2-3-4-5

2-3-4-5-6

......

10-J-Q-K-A

So the probability  is just  

C(10,1) * C(4,1)/ C(52,5)  ≈  : .00394  = .394%

(a) What is the probability that we draw 1 ace, 1 two, 1 three, 1 four, and 1 five (this is one way to get a "straight")?

In each case  we want to  select  1 of 4 cards from each of the specified ranks  = [ C(4,1)]^5  = 4^5

And the total  hands = C(52,5)

So.....the probability is just

4^5 / C(52,5)   ≈  .000394  = .0394%

Note that these hands include the possibility of a straight flush ....

LOL!!!!

Thx, Guest......

Checking to see that a solution [ pardon the pun ]  is "reasonable" is always a good thing  !!!

Thx,  Guest.....your procedure is WAY more elegant than mine

But....we  both got to the same destination

You took a jet

I took Greyhound