If k < ( 2 +1)^3 < k+2 , where k is a natural number, then what is the value of k ?

Step-by-step solution

Let x = ( 2 +1)^3 . Let x = I + f , where I is the integer part and 0 Let f' = ( 2 -1)^3 . Since 0 Now, evaluate x - f' : x - f' = ( 2 +1)^3 - ( 2 -1)^3 x - f' = (2 2 + 6 + 3 2 + 1) - (2 2 - 6 + 3 2 - 1) x - f' = (7 + 5 2 ) - (5 2 - 7) = 14 Since x = I + f , we get I + f - f' = 14 , which implies f - f' = 14 - I . Since 14 - I is an integer and -1 Thus, f = f' and I = 14 . This gives x = 14 + f , meaning 14 We are given the inequality k Checking the given options: For k = 11 , the interval is (11, 13) , which does