***urgent*** ***50 points***
if you do answer, provide an explanation because i want to know how to solve these problems for the future and not just have the answer without an explanation.

27) 18 < P ≤ 18 + 6√2 ⇒ answer D

28) The sum of the degree measures these angles is 1080° ⇒ answer B

29) 3E minutes before A ⇒ answer B

30) The difference between the greatest possible values is 0 ⇒ answer E

31) r divided by s = 1/3 ⇒ answer A

Step-by-step explanation:

* Lets explain each problem

27)

∵ BE is a quarter circle

∵ The radius of the circle is 6

∵ Point c is on the arc BE

∴ The distance from D to C = 6 ⇒ not depends on the position of c

   because DC is a radius in the quarter circle BE

- In Δ BDE

∵ m∠ D = 90°

∵ DB = DE = 6 ⇒ radii of the quarter circle

- By using Pythagoras Theorem

∴ BE = √ (6² + 6²) = √(36 + 36) = √72 = 6√2

- The perimeter of the quadrilateral ABCD is the sum of the sides

∵ AB = 6 , AD = 6 , CD = 6

- Point C can move from B to E

∴ The length of side BC can b greater than 0(it can not be 0

   because the quadrilateral has 4 sides

∴ The length of BC can not exceed the length of BE because the last

   position of point C to be on the arc BE is point E

∴ The length of BC ⇒ 0 < BC ≤ 6√2

  equal 6√2

∵ P is the perimeter of the quadrilateral ABCD

∴ P = 6 + 6 + 6 + (0 < BC ≤ 6√)

∴ P = 18 + (0 < BC ≤ 6√)

- Add 18 to 0 and 18 to 6√2

∴ 18 < P ≤ 18 + 6√2

28)

- In the figure we have a quadrilateral

- All the arrows represent the exterior angles of the figures

- Use the fact that:

 The sum of all angles around a points is 360°

∵ There are 4 vertices (points) on the quadrilateral

∴ The sum of the all angles around the 4 vertices = 4 × 360 = 1440°

- Use the fact that:

 The sum of the interior angles of any quadrilateral is 360°

∵ The sum of the angles represented by the arrows is the difference

  between the sum of all angles around the 4 vertices and the sum

  of the interior angles of the quadrilateral

∴ The sum of these angles = 1440° - 360° = 1080°

* The sum of the degree measures these angles is 1080°

29)

- In any watch the short arrow-hand represents the hours and the long

 arrow-hand represents the minutes

- The numbers of the hours in the watch from 1 to 12

- The number of minutes between each two hours is 5 minutes, then

  at 1 o'clock the minutes number is 5 , at 6 o'clock the number of

  minutes is 30 , at 9 o'clock the number of minutes is 45 , so we can

  find the number of minutes at any number of hour by multiply the

  number of hour by 5

∵ The number of hours have been replaced by letters

∵ The time on the watch is 45 minutes after 12 o'clock OR

  15 minutes before 1 o'clock

∵ The short arrow-hand pointed between L and A

∵ L is the replacing of 12 o'clock and A is the replacing of 1 o'clock

∵ The long arrow-hand pointed at I

∵ I is the replacing of  9 o'clock

∵ The hour number 9 means 5 × 9 = 45 minutes

∴ The hour hand I has 5I minutes

∴ The time in letter is 5I minutes after L

- This answer is not in the choices

- But the answer of 3E minutes before A means:

∵ E is the replacing of 5 o'clock

∴ 3E = 3 × 5 = 15 minutes

∵ A is the replacing of 1 o'clock

∴ 3E minutes before A means 15 minutes before 1 o'clok

* The answer is ⇒ 3E minutes before A

30)

∵ r² = 9

∴ r = ± √9 = ± 3

∴ r has two values -3 and 3

∵ s² = 25

∴ s = ± √25 = ± 5

∴ s has two values -5 and 5

- To find the greatest value of s - r put s greatest and r smallest

∵ The greatest value of s is 5

∵ The smallest value of r is -3

∴ The greatest value of s - r = 5 - (-3) = 5 + 3 = 8

- To find the greatest value of r - s put r greatest and s smallest

∵ The greatest value of r is 3

∵ The smallest value of s is -5

∴ The greatest value of r - s = 3 - (-5) = 3 + 5 = 8

∴ The difference between the greatest possible values of s - r

   and r - s = 8 - 8 = 0

* The difference between the greatest possible values is 0

31)

- There are 27 cubes each of side length r

- The 27 cubes are arranged to form on single large cube of side

  length s

∵ The volume of any cube is V = L³ , where L is the length of its side

∵ The large cube formed from the 27 small cubes

∴ The volume of the large cube = the volume of the 27 small cubes

∵ The side of the small cube is r

∴ The volume of the small cube is r³

∵ The side of the large cube is s

∴ The volume of the large cube is s³

∴ s³ = 27 r³

- Divide both sides by s³ and 27

∴ s³/(27 s³) = (27 r³)/(27 s³)

∴ 1/27 = r³/s³

- Take ∛  for both sides

∴ ∛(r³/s³) = ∛(1/27)

- The cube root canceled by the power 3 and the cube root of

  1/27 is 1/3

∴ r/s = 1/3

* r divided by s = 1/3