A vertical-launch research drone is programmed to reach a target height of 500 metres. The drone’s vertical velocity, w m s−1w\text{ m s}^{-1}w m s−1, is modelled by the equation:
w=25.6−20e−0.5t−0.04e0.25t w = 25.6 - 20e^{-0.5t} - 0.04e^{0.25t} w=25.6−20e−0.5t−0.04e0.25twhere ttt is the time in seconds after the start of the ascent phase.
Find the maximum vertical velocity of the drone, giving your answer to one decimal place. Fully justify that the value you have found is a maximum.
Find an expression for the height of the drone above its starting position, hhh, in terms of ttt.
In a specific test flight, the drone reached the 500-metre target in exactly 28.0 seconds. Comment on the accuracy of the model.
Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of 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.
