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 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.