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 Differentiation with exam-style questions for A Level Maths. 311 questions covering 9.1 Differentiating sin x and cos x, 9.2 Differentiating exponentials and logarithms, 9.3 The Chain Rule, 9.4 The Product Rule, 9.5 The Quotient Rule, 9.6 Differentiating Trigonometric Functions, 9.7 Parametric Differentiation, 9.8 Implicit Differentiation, 9.9 Using Second Derivatives, and 9.10 Rates of Change, 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.
