CPhill

Note that the radius of the largest circle is  OC + CB = 32 + 36 =  68

The radius of the smallest circle is (1/2)CB = (1/2)(36) = 18

The radius of the intermediate circle is the tricky one to figure !!!

Note that AO is the radius of the largest circle = 68

And OC = 32

So....AO + OC = the diameter of the intermediate circle = 68 + 32 = 100

So....the radius of this circle = 50

So....the shded area is

Area of largest circle - area of the other two circles =

pi  * [ 68^2 - (50^2 + 18^2) ]  =

pi  * [ 4624 - ( 2500 + 324) ]   =

pi * [ 4624 - 2824]  =

pi * [ 1800] units^2   ≈  5654.87 units^2

Here's a graphical solution for the second one :

https://www.desmos.com/calculator/fnueaahoif

If we let  x be the number of pies and y be the number of batches of brownies.....we can graph the inequality 

8x + 12y ≤ 112

Lookng at the graph....we can forget the points inside the shaded area....these combinations will not yield enough desserts

Note that the red dot - representing 3 pies and 8 batches of brownies - is closer to the boundary of the inequality than the points (2, 9)   and (5,7)

So   

(3, 8)  yields = 120 desserts

(2, 9) yields = 124 desserts

(5, 7) yields = 124 desserts

So.....we want to choose   3 pies and 8 batches of brownies

Thanks to EP  for spotting my earlier error  !!!

I think it is because I didn't consider the space between the barrels....

Notice that......if we had 1 barrel....the actual area occupied by the barrel is a  2.5 x 2.5 square  =

6.25 ft^2

So    52.5 / 6.25  = 8.4 barrels....( round to 8 barrels)

To get a feel for this....look at the following image :

That's an interesting way of solving this problem, heureka......how was that method derived???

We look at the  digit after the 5.   Since it is a "9," we round the "5" to a "6"

So we have : 46,000

Correct, EP   !!!

18 = 6 * 3  =  2* 3 * 3

93 = 31 * 3

31 and 2  are co-prime.....so.....GCF =  3

The lines 2x+y=k and 0.5x+y=14 intersect when x=  -8.4. What is the value of k?

The intersection point will also have the same y value

So

-2x + k   =     -0.5x + 14        

-2(-8.4) + k = -.0.5(-8.4) + 14

16.8 + k =  4.2 + 14

16.8 + k =  18.2

k = 18.2 - 16.8   =   1.4

Looks like the difference between terms is  -48.8 -  (-35) = -48.8 + 35 =    -13.8

So.......

a_n = a_n-1 - 13.8 , where a₁ = - 7.4

 [ Remember that  a_n  is the term after a_n-1 ]

B

6          10

A           C

Cos A  =  AC / BC

By the Pythagorean Theorem

AC = sqrt ( BC^2 - AB^2)  = sqrt ( 10^2 - 6^2)  = sqrt (100 - 36)   = sqrt (64) = 8

So

Cos A =    8 / 10   =  4 / 5