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1 Pure Mathematics

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Question 464

An amphitheatre stage is designed as a composite of two circular sectors sharing a common centre OOO. The outer part of the stage is a minor sector POQ POQ\,POQ with radius OP OP\,OP and arc PQPQPQ. The inner part is a reflex (major) sector ROS ROS\,ROS with radius OR OR\,OR and arc RTSRTSRTS. It is given that points R R\,R and S S\,S lie on the radii OP OP\,OP and OQ OQ\,OQ respectively, such that O,R,P O, R, P\,O,R,P and O,S,Q O, S, Q\,O,S,Q are collinear.

The layout specifications are as follows:

  • The central angle ∠POQ \angle POQ\,∠POQ of the minor sector is 0.8 radians.
  • The length of the outer arc PQ PQ\,PQ is 20 m.
  • The ratio of the lengths PR:RO=1:4PR : RO = 1 : 4PR:RO=1:4.
a.

Show that RO=20 mRO = 20\text{ m}RO=20 m.

[4]
b.

Calculate the total perimeter of the amphitheatre stage, giving your answer in m to 3 significant figures.

[5]
c.

Calculate the total surface area of the stage, giving your answer in m2\text{m}^2m2 to the nearest integer.

[5]
Markscheme

1 Pure Mathematics Questions

  1. A Level
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