A circular safety valve with radius 8 cm is designed to open gradually. The cross-section of the opening is a segment of a circle with centre O O\,O and radius rrr. The angle subtended by the segment at the centre is θ \theta\,θ radians.
Given that:
Show that dAdθ=K(1−cosθ)\frac{dA}{d\theta} = K(1 - \cos \theta)dθdA=K(1−cosθ) where K K\,K is a constant to be found.
Find, in cm2 ^2\,2 s−1^{-1}−1, the rate of increase of the area of the valve opening when θ=π4\displaystyle \theta = \frac{\pi}{4}θ=4π. Give your answer in the form a+b2a + b\sqrt{2}a+b2, where a a\,a and b b\,b are integers.
Practise Edexcel A Level Maths 9.10 Rates of Change with exam-style questions for A Level Maths. 24 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.