A mechanical diaphragm for a high-precision camera lens controls light by opening a circular segment-shaped aperture. The aperture has a fixed radius of 4 cm and is defined by the central angle θ\thetaθ radians. As the shutter opens, θ\thetaθ increases at a constant rate of 0.4 radians per second. The area of the opening is A cm2A \text{ cm}^2A cm2.
Show that
dAdθ=K(1−cosθ) \frac{dA}{d\theta} = K(1 - \cos \theta) dθdA=K(1−cosθ)where KKK is a constant to be found.
Find the rate at which the area of the aperture is increasing when θ=3π4\theta = \frac{3\pi}{4}θ=43π. Give your answer in the form a+b2 cm2s−1a + b\sqrt{2} \text{ cm}^2\text{s}^{-1}a+b2 cm2s−1, where aaa and bbb are constants.
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.
