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.
717 exam-style questions on Edexcel A Level Maths Differentiation, 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. Each one has a worked solution and a mark scheme showing where the marks go.