O-Level Surds: worked solution

Question

A closed cylinder of volume (66√2 + 2√3)π cm³ has radius (√2 + 2√3) cm and height h cm. Write h in the form p√2 + q√3 for integers p, q.

Method: use the cylinder volume formula V = πr²h, make h the subject, square the surd radius using (a + b)², then rationalise the denominator with its conjugate. Step 1 Volume of a cylinder. V = πr²h Divide both sides by πr²: h = V/(πr²) Step 2 Substitute V = (66√2 + 2√3)π and r = √2 + 2√3. The π on top cancels the π underneath. h = (66√2 + 2√3)π / [π(√2 + 2√3)²] h = (66√2 + 2√3)/(√2 + 2√3)² Step 3 Square the radius, using (a + b)² = a² + 2ab + b² with a = √2 and b = 2√3. a² = (√2)² = 2 2ab = 2 × √2 × 2√3 = 4√6 b² = (2√3)² = 4 × 3 = 12 (√2 + 2√3)² = 2 + 4√6 + 12 = 14 + 4√6 Step 4 So h = (66√2 + 2√3)/(14 + 4√6) Rationalise by multiplying top and bottom by the conjugate 14 − 4√6. Denominator, using (a + b)(a − b) = a² − b²: (14 + 4√6)(14 − 4√6) = 14² − (4√6)² = 196 − 16 × 6 = 196 − 96 = 100 Step 5 Numerator: expand (66√2 + 2√3)(14 − 4√6) term by term. 66√2 × 14 = 924√2 66√2 × (−4√6) = −264√12 = −264 × 2√3 = −528√3 (since √12 = √4 × √3 = 2√3) 2√3 × 14 = 28√3 2√3 × (−4√6) = −8√18 = −8 × 3√2 = −24√2 (since √18 = √9 × √2 = 3√2) Step 6 Collect the like surds. √2 terms: 924√2 − 24√2 = 900√2 √3 terms: −528√3 + 28√3 = −500√3 Numerator = 900√2 − 500√3 Step 7 Divide by the denominator 100. h = (900√2 − 500√3)/100 = 9√2 − 5√3 So h = 9√2 − 5√3 cm, that is p = 9 and q = −5. (As a check, 9(1.41421) − 5(1.73205) = 12.728 − 8.660 = 4.07 cm, a sensible positive height.)