CPhill

Very nice, jugoslav  !!!!

Let's see how the guest derived his/her answer.....

Look at the following:

The area we seek  is (1/2)  the area of the octagon  less the area of  triangle CDE

Note that angle CDE = 135°

Area of triangle  CDE =  (1/2) (CD) (DE) sin (135)  = (1/2) (12) (12) ( 1/ sqrt(2))  = 72/sqrt (2)  = 36sqrt (2)   [1]

Area of (1/2) of the octagon =  ( 1 + sqrt (2) ) * side^2  =  (1 + sqrt (2) ) (12)^2  =  (1 + sqrt (2) ) (144)  =

144 + 144sqrt (2)     [2]

Area of trapezoid =     [2]  - [1]   =    [ 144 + 108 sqrt (2)] cm^2  ≈  296.735 cm^2

Look at the graph here, Thanks4Helping :  https://www.desmos.com/calculator/pxlandri3i

The over-lapping areas of the  two  inequalities represent the  feasible region....(it will appear as a grey area)

Note that we can immediately rule out any  points  lying  below the x axis  [ points  B, F, J , H  and D ]

E = (5,6)   not in  feasible region

A = (3,3)   not in feasible region

I = (2,4)    this point is on the border of the solid line....it is included in the region

G= (-3, 5)  in the feasible region

C = (-4,1) not in feasible  region

So....the  points are   ( I , G )

Thx, jugoslav  !!!!

Thx, guest....here's another way

If AB  passes through  (0,0)  and has a slope of 2/3, the equation of a line through AB  is

y= (2/3)x

Since  B is on  the line y =5, then  the y coordinate of B  = 5

To  find the  x coordinate we have

5 = (2/3) x    multiply both sides by 3/2

(3/2) 5  = x  =  7.5

B= (7.5 ,5)   and the sum of the coordinates  =  12.5

64p - 144 ≤  0  →   64p ≤ 144   →  p ≤ 144/64  →   p ≤  9/4

22 - 20p < 0  →   22 < 20p   →  22/20 < p  →   p >  11/10

p = ( 11 / 10,  9 / 4 ] 

Thx....glad to help  !!!!

Total capacity =  pi * r^2 * h  / 3   =    pi * (16)^2 * 120  / 3 =  10240  pi ft^3

.25 capacity = 2560 pi

Using similar  triangles we  have that        r / h =  16 /120  =  2 /15  so    r =  2h  /15   

So

2560 pi  =  pi  ( 2h/15)^2  * h    simplify

2560  = (4/225)h^3

2560 (225/4)  = h^3

144000 = h^3       take the cube root of both sides

 h ≈ 52.4 ft   when  25%  of capacity

2) is  a little  difficult

At the end of Week 5 of my basketball team's season, our record is 3 wins and 17 losses. Each week after that, we win 3 games and lose 1 game. At the end of which week will my team have won at least 65% of its games total for the first time in the season?

Call the  number of  additional weeks  (past week 5) that it takes to win 65% of the games,  W

Our team plays 4 games per week and we win 3 of these

So

Total games won

________________  =     .65

Total games played

3  + 3 W

____________  =   .65

(17 + 3) + 4W

3 + 3 W

________  = .65           multiply both sides  by  20 + 4W

20 + 4W

3 + 3 W  =  .65 ( 20 + 4 W )     simplify

3 + 3 W  = 13  + 2.6 W      subtract  2.6 W, 3   from both sides

.4W  = 10        divide both sides by .4

W = 25   (additional weeks past week 5 )   ....so  ...after the 30th week

Check

3 + 25(3)                   78

__________   =      ____  =  .65    !!!!

20 + 25 (4)              120

I'll assume  that this is   10t / { 25 + t ]

So

3.75  = 10t  / [ 25 + t]        multiply both sides by  25 + t

3.75 [ 25 + t]  = 10t      simplify

93.75  + 3.75 t =10 t      subtract 3.75 t from both sides

93.75  = 6.25 t     divide both sides by 6.25

t  =15  (min)