An automated deep-sea submersible is programmed to perform vertical transects for water sampling. Its depth, HHH metres below the surface, is recorded at various times ttt hours after its deployment at midnight. The data points collected are used to create the following model:
H=7.4cos(2π(t−3.2)18)+12.5 H = 7.4 \cos\left( \frac{2\pi(t - 3.2)}{18} \right) + 12.5 H=7.4cos(182π(t−3.2))+12.5Find the minimum depth below the surface reached by the submersible, according to the model. Give your answer to the nearest centimetre.
Find the duration of the longest continuous period during the first day (from t=0t=0t=0 to t=24t=24t=24) where the depth of the submersible is predicted to be less than 8 metres. Give your answer in hours to one decimal place.
A different model is needed for a second submersible exploring a deeper trench. This second submersible exhibits a more stable vertical movement (a smaller vertical range) around the same average depth. A technician suggests refining the existing model for this second submersible by increasing the value of 7.4 to 8.2. Explain whether this refinement is appropriate.
317 exam-style questions on AQA A Level Maths 1.8 E: Trigonometry, covering 1.8.1 Trigonometric definitions, rules and radians, 1.8.2 Small angle approximations (A-level only), 1.8.3 Trigonometric functions and exact values, 1.8.4 Reciprocal and inverse trigonometric functions (A-level only), 1.8.5 Trigonometric identities, 1.8.6 Compound and double angle formulae (A-level only), 1.8.7 Solving trigonometric equations, 1.8.8 Proofs with trigonometric identities, and 1.8.9 Trigonometry in context. Each one has a worked solution and a mark scheme showing where the marks go.