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.
375 exam-style questions on AQA A Level Maths 1.10 G: Differentiation, covering 1.10.1 The derivative and second derivative, 1.10.2 Differentiating standard functions, 1.10.3 Applications of differentiation, 1.10.4 Product, quotient and chain rules (A-level only), 1.10.5 Implicit and parametric differentiation (A-level only), and 1.10.6 Constructing differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.