The curved surface area of cylinder P is 16632 cm²  whose height and radius are (y+2) and (z-7) respectively. The radius of cylinder Q is 14 cm more than the radius of cylinder P. The curved surface area of cylinder P is 792 cm²  less than that of cylinder Q. If the height of cylinder Q is equal to the square of six, then which of the following statements is/are true?

(i) The value of ‘y’ is the multiple of 7.

(ii) The ratio between the volume of cylinder P and Q is (5y-11) : (3z+32) respectively.

(iii) The value of [(difference between the values of ‘y’ and ‘z’)/10] is a prime number.

Solution

The curved surface area of cylinder P is 16632 cm²  whose height and radius are (y+2) and (z-7) respectively. curved surface area of cylinder P = 2 x (22/7) x (z-7) x (y+2) = 16632 (44/7) x (z-7) x (y+2) = 16632 (1/7) x (z-7) x (y+2) = 378 (z-7) x (y+2) = 2646    Eq.(i) The radius of cylinder Q is 14 cm more than the radius of cylinder P. radius of cylinder Q = (z-7)+14 = (z+7) If the height of cylinder Q is equal to the square of six height of cylinder Q = 6x6 = 36 cm The curved surface area of cylinder P is 792 cm²  less than that of cylinder Q. The curved surface area of cylinder P is 16632 cm² . 16632 = (curved surface area of cylinder Q) - 792 curved surface area of cylinder Q = 16632+792 = 17424 cm² 2 x (22/7) x (z+7) x 36 = 17424 (44/7) x (z+7) x 36 = 17424 (1/7) x (z+7) = 11 (z+7) = 77 z = 77-7 z = 70 Put the value of ‘z’ in Eq.(i). (70-7) x (y+2) = 2646 63 x (y+2) = 2646 (y+2) = 42 y = 42-2 y = 40 (i) The value of ‘y’ is the multiple of 7. The above given statement is not true. Because the value of ‘y’ is not the multiple of 7. (ii) The ratio between the volume of cylinder P and Q is (5y-11) : (3z+32) respectively. the ratio between the volume of cylinder P and Q = (5y-11) : (3z+32) = (5x40-11) : (3x70+32) = (200-11) : (210+32) = 189 : 242 So the volume of cylinder P and Q = [22/7 x (z-7) ² x (y+2)] : [22/7 x (z+7) ² x 36]  Put the value of  ‘y’ and ‘z’. = [22/7 x (z-7) ² x (y+2)] : [22/7 x (z+7) ² x 36]  = [(70-7) ² x (40+2)] : [(70+7) ² x 36] = [(63) ² x 42] : [(77) ² x 36] = [63 x 63 x 42] : [77 x 77 x 36] = [9 x 9 x 7] : [11 x 11 x 6] = [9 x 3 x 7] : [11 x 11 x 2] = 189 : 242 The above given statement is true. (iii) The value of [(difference between the values of ‘y’ and ‘z’)/10] is a prime number. [(z-y)/10] = prime number [(70-40)/10] = prime number [30/10] = prime number prime number = 3 The above given statement is true.