A specialized coolant storage tank has a depth of 20 cm20\text{ cm}20 cm. The tank is initially empty and a liquid refrigerant is pumped into it. When the depth of the refrigerant is h cmh\text{ cm}h cm, the volume of the liquid in the tank, V cm3V\text{ cm}^3V cm3, is modelled by the equation
V=15h2(h+15)0≤h≤20 V = \frac{1}{5}h^2(h + 15) \quad 0 \le h \le 20 V=51h2(h+15)0≤h≤20The refrigerant is pumped into the tank at a constant rate of 350 cm3 s−1350\text{ cm}^3\text{ s}^{-1}350 cm3 s−1. According to the model:
calculate the time taken to fill the tank to its maximum depth.
determine the rate of change of the depth of the liquid, in cm s−1\text{cm s}^{-1}cm s−1, at the instant when h=10h = 10h=10.
375 exam-style questions on AQA A Level Maths 1.10 G: Differentiation, 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). Each one has a worked solution and a mark scheme showing where the marks go.