In the diagram, ASRTB represents a piece of string passing over a pulley of radius 10cm in a vertical plane. O is the centre of the pulley and AMB is a horizontal straight line touching the pulley at M. Angle SAB = 90^\circ and angle TBA = 60^\circ.
(a) Calculate (i) the obtuse angle SOT ; (ii) arc SRT ; (iii) |BT|
(b) Find, correct to the nearest cm, the length of the string. (Take \(\pi = \frac{22}{7}\)).
Verified Correct Answer: .
Step-by-Step Mathematical Solution
(a)(i) < ASO = < AMO = 90^\circ (radius to a tangent)
< SOM = 360^\circ - (3 x 90^\circ) (sum of angles of a rectangle)
= 90^\circ.
\(\therefore\) obtuse < SOT = 360^\circ - (90^\circ + 60^\circ + 60^\circ) = 150^\circ.
(ii) Arc SRT = \(\frac{150^\circ}{360^\circ} \times 2\pi r\)
= \(\frac{150}{360} \times 2 \times \frac{22}{7} \times 10 = \frac{550}{21}\)
= \(26.19cm\)
(iii) \(\tan 60^\circ = \frac{TB}{10}\)
\(TB = 10\tan 60^\circ = 17.32cm\)
(b) Length of the string ASRTB = AS + arc SRT + TB
= 10cm + 26.19cm + 17.32cm
= 53.51cm \(\approxeq\) 54cm.
Source: Myschool Past Questions Archive (Record #54453)