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.
115 exam-style questions on CCEA A Level Maths 1.6 Differentiation, covering 1.6.1 Differentiation, 1.6.2 Differentiation, 1.6.3 Differentiation, 1.6.4 Differentiation, 1.6.5 Differentiation, 1.6.6 Differentiation, and 1.6.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.