A bicycle repair shop calculates a stress ratio for a welded frame joint during a routine annual safety inspection carried out on every bike left overnight,...
Question 1 Report
A bicycle repair shop calculates a stress ratio for a welded frame joint during a routine annual safety inspection carried out on every bike left overnight, and the value works out as \(\frac{5}{2 - \sqrt{3}}\) in the engineer's notes.
- Rationalise the denominator, giving your answer in the form \(a + b\sqrt{3}\). (3)
- Hence state whether the ratio is greater than 9. (1)
Answer Details
A denominator containing \(a-\sqrt b\) is rationalised by multiplying top and bottom by its conjugate \(a+\sqrt b\), since \((a-\sqrt b)(a+\sqrt b)=a^2-b\) has no surd left in it.
- Multiplying by \(\dfrac{2+\sqrt3}{2+\sqrt3}\) [1 mark]: the denominator becomes \((2-\sqrt3)(2+\sqrt3)=2^2-(\sqrt3)^2=4-3=1\) [1 mark]; the numerator becomes \(5(2+\sqrt3)=10+5\sqrt3\). Since the denominator is \(1\), the ratio simplifies to \(10+5\sqrt3\), matching the form \(a+b\sqrt3\) with \(a=10\), \(b=5\). [1 mark]
- Since \(\sqrt3>1\), it follows that \(5\sqrt3>5\), so \(10+5\sqrt3>10+5=15\), which is already greater than \(9\); the ratio is greater than \(9\). [1 mark]
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