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 AQA A Level Maths 1.10 G: Differentiation with exam-style questions for A Level Maths. 321 questions 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), matched to the AQA A Level Maths (7357) 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.
