***URGENT*** ***50 POINTS***If you do answer, please 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.

Answer:27) 18 < P ≤ 18 + 6√2 ⇒ answer D28) The sum of the degree measures these angles is 1080° ⇒ answer B29) 3E minutes before A ⇒ answer B30) The difference between the greatest possible values is 0 ⇒ answer E31) r divided by s = 1/3 ⇒ answer AStep-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√228) - 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 A30)∵ 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 031)- 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