Calculate, correct to three significant figures, the length of the arc AB in the diagram above.
[Take \(\pi = ^{22}/_7\)]
A.
32.4 cm
B.
30.6 cm
C.
28.8 cm
D.
30.5 cm
Correct answer is B
Consider ∆XOB and using Pythagoras theorem
13\(^2\) = 12\(^2\) + h\(^2\)
⇒ 169 = 144 + h\(^2\)
⇒ 169 - 144 =h\(^2\)
⇒ 25 = h\(^2\)
⇒ h = \(\sqrt25\) = 5cm
tan θ = \(\frac {opp}{adj}\)
⇒ tan θ = \(\frac{12}{5}\) = 2.4
⇒ θ = tan\(^{-1}\)(2.4)
⇒ θ = 67.38\(^0\)
∠AOB = 2θ = 2 x 67.38\(^o\) = 134.76\(^o\)
L = \(\frac{θ}{360^o} \times 2\pi r\)
⇒ L = \(\frac {134.76}{360} \times 2 \times \frac {22}{7} \times 13 = \frac {77082.72}{2520}\)
∴ L = 30.6cm (to 3 s.f)