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

(√2 + 1)³ = (√2)³ + 3(√2)²(1) + 3(√2)(1)² + 1 = 2√2 + 6 + 3√2 + 1 = 7 + 5√2. Now 5√2 ≈ 5·1.4142 = 7.071, so (√2+1)³ ≈ 14.071. We need k < 14.071 < k + 2, so k can be 13 (13 < 14.071 < 15) or 12. Among the options 13 fits.